Differentiation

Differentiation is a method of finding rates of change, i.e. the gradients of functions. The result of differentiating a function is called the **derivative** of that function.

Explore our app and discover over 50 million learning materials for free.

- Applied Mathematics
- Calculus
- Decision Maths
- Discrete Mathematics
- Geometry
- Logic and Functions
- Mechanics Maths
- Probability and Statistics
- Pure Maths
- ASA Theorem
- Absolute Convergence
- Absolute Value Equations and Inequalities
- Abstract algebra
- Addition and Multiplication of series
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebra of limits
- Algebra over a field
- Algebraic Fractions
- Algebraic K-theory
- Algebraic Notation
- Algebraic Representation
- Algebraic curves
- Algebraic geometry
- Algebraic number theory
- Algebraic topology
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Associative algebra
- Average Rate of Change
- Banach algebras
- Basis
- Bijective Functions
- Bilinear forms
- Binomial Expansion
- Binomial Theorem
- Bounded Sequence
- C*-algebras
- Category theory
- Cauchy Sequence
- Cayley Hamilton Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Clifford algebras
- Cohomology theory
- Combinatorics
- Common Factors
- Common Multiples
- Commutative algebra
- Compact Set
- Completing the Square
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Congruence Equations
- Conic Sections
- Connected Set
- Construction and Loci
- Continuity and Uniform convergence
- Continuity of derivative
- Continuity of real valued functions
- Continuous Function
- Convergent Sequence
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Coupled First-order Differential Equations
- Cubic Function Graph
- Data Transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Derivative of a real function
- Deriving Equations
- Determinant Of Inverse Matrix
- Determinant of Matrix
- Determinants
- Diagonalising Matrix
- Differentiability of real valued functions
- Differential Equations
- Differential algebra
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Dimension
- Direct and Inverse proportions
- Discontinuity
- Disjoint and Overlapping Events
- Disproof By Counterexample
- Distance from a Point to a Line
- Divergent Sequence
- Divisibility Tests
- Division algebras
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Eigenvalues and Eigenvectors
- Ellipse
- Elliptic curves
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Equicontinuous families of functions
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Fermat's Little Theorem
- Field theory
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding The Area
- First Fundamental Theorem
- First-order Differential Equations
- Forms of Quadratic Functions
- Fourier analysis
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Gram-Schmidt Process
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs And Differentiation
- Graphs Of Exponents And Logarithms
- Graphs of Common Functions
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Grothendieck topologies
- Group Mathematics
- Group representations
- Growth and Decay
- Growth of Functions
- Gröbner bases
- Harmonic Motion
- Hermitian algebra
- Higher Derivatives
- Highest Common Factor
- Homogeneous System of Equations
- Homological algebra
- Homotopy theory
- Hopf algebras
- Hyperbolas
- Ideal theory
- Imaginary Unit And Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Injective linear transformation
- Instantaneous Rate of Change
- Integers
- Integrating Ex And 1x
- Integrating Polynomials
- Integrating Trigonometric Functions
- Integration
- Integration By Parts
- Integration By Substitution
- Integration Using Partial Fractions
- Integration of Hyperbolic Functions
- Interest
- Invariant Points
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Inverse of a Matrix and System of Linear equation
- Invertible linear transformation
- Iterative Methods
- Jordan algebras
- Knot theory
- L'hopitals Rule
- Lattice theory
- Law Of Cosines In Algebra
- Law Of Sines In Algebra
- Laws of Logs
- Leibnitz's Theorem
- Lie algebras
- Lie groups
- Limits of Accuracy
- Linear Algebra
- Linear Combination
- Linear Expressions
- Linear Independence
- Linear Systems
- Linear Transformation
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition And Subtraction
- Matrix Calculations
- Matrix Determinant
- Matrix Multiplication
- Matrix operations
- Mean value theorem
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modelling with First-order Differential Equations
- Modular Arithmetic
- Module theory
- Modulus Functions
- Modulus and Phase
- Monoidal categories
- Monotonic Function
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplicative ideal theory
- Multiplying And Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Non-associative algebra
- Normed spaces
- Notation
- Number
- Number Line
- Number Systems
- Number Theory
- Number e
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations With Matrices
- Operations with Decimals
- Operations with Polynomials
- Operator algebras
- Order of Operations
- Orthogonal groups
- Orthogonality
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Hyperbolas
- Parametric Integration
- Parametric Parabolas
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Pointwise convergence
- Poisson algebras
- Polynomial Graphs
- Polynomial rings
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Determinants
- Properties of Exponents
- Properties of Riemann Integral
- Properties of dimension
- Properties of eigenvalues and eigenvectors
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic forms
- Quadratic functions
- Quadrilaterals
- Quantum groups
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Ratio and Root test
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Rearrangement
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Reduced Row Echelon Form
- Reducible Differential Equations
- Remainder and Factor Theorems
- Representation Of Complex Numbers
- Representation theory
- Rewriting Formulas and Equations
- Riemann integral for step function
- Riemann surfaces
- Riemannian geometry
- Ring theory
- Roots Of Unity
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Products
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Fundamental Theorem
- Second Order Recurrence Relation
- Second-order Differential Equations
- Sector of a Circle
- Segment of a Circle
- Sequence and series of real valued functions
- Sequence of Real Numbers
- Sequences
- Sequences and Series
- Series Maths
- Series of non negative terms
- Series of real numbers
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Similarity and diagonalisation
- Simple Interest
- Simple algebras
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Spanning Set
- Special Products
- Special Sequences
- Standard Form
- Standard Integrals
- Standard Unit
- Stone Weierstrass theorem
- Straight Line Graphs
- Subgroup
- Subsequence
- Subspace
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Summation by Parts
- Supremum and Infimum
- Surds
- Surjective functions
- Surjective linear transformation
- System of Linear Equations
- Tables and Graphs
- Tangent of a Circle
- Taylor theorem
- The Quadratic Formula and the Discriminant
- Topological groups
- Torsion theories
- Transformations
- Transformations of Graphs
- Transformations of Roots
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Uniform convergence
- Unit Circle
- Units
- Universal algebra
- Upper and Lower Bounds
- Valuation theory
- Variables in Algebra
- Vector Notation
- Vector Space
- Vector spaces
- Vectors
- Verifying Trigonometric Identities
- Volumes of Revolution
- Von Neumann algebras
- Writing Equations
- Writing Linear Equations
- Zariski topology
- Statistics
- Theoretical and Mathematical Physics

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenDifferentiation is a method of finding rates of change, i.e. the gradients of functions. The result of differentiating a function is called the **derivative** of that function.

A **differential equation** is a type of equation that involves derivatives. In other words, a differential equation represents a situation where** the rate of change of a quantity is dependent on the current state of the quantity**.

Differential equations are broadly classified into two types:

**Ordinary Differential Equations (ODEs)**: These involve derivatives with respect to**only one variable**. They are further classified based on the order (the highest derivative in the equation) and the degree (the power of the highest derivative in the equation).**Partial Differential Equations (PDEs)**: These involve derivatives with respect to**more than one variable**.

Differential equations are typically represented symbolically. For example, an ordinary differential equation might be written as: \[f'(x) = \frac{dy}{dx}\]

This is equivalent to 'change in y divided by change in x'. Variables x and y can be substituted for any other letter.

In the case of partial derivatives, the "d"s would be instead represented by the ∂ symbol: \(f'(x) = \frac{∂y}{∂x}\).

The apostrophe (') written behind the letter symbolising a function denotes that the equation that follows is not the original equation but its derivative. Differentiating a function y 'with respect to x' (meaning x is the value on the bottom of the fraction) results in the derivative y '. If the function is represented as f (x), then its derivative can be represented as f'(x).

Let's do a quick review of how to find the gradient of a straight line graph:

However, if we look at a quadratic graph, it isn't clear how to find its gradient. This is because it changes at different points in the graph as the line curves, getting more or less steep.

One potential method we could use is to draw a tangent at a given point and find its equation. However, this would only give us the gradient at that point - what if we wanted to find a general expression for the gradient of any point on the graph?

We use differentiation to find a function for the gradient of a graph. The method is very straightforward - you need to:

Decrease the power of x by one

Multiply by the old power

Therefore, as a general rule, when differentiating x^{n}, your result is \(nx^{n-1}\).

Let's say we have the following graph of \(y = x^2 + x+2\) and we want to find the gradient at the point x = 1.

To differentiate the function, we take each power of x and perform the above steps on it - reduce the power by 1, and multiply by the old power.

\(y = x^2 + x +2\)

\(x^2 \Rightarrow 2x^{2-1} = 2x\)

\(x \Rightarrow x^{1-1} = x^0 = 1\)

2 isn't a power of x, so we can't apply our usual method here.

To understand how to differentiate it, we need to look at the representation of differentiation \(\frac{dy}{dx}\). As a reminder, this means 'the change in y divided by the change in x'.

Since 2 is a constant, changes in x and y do not affect its value, and vice versa. This effectively means that for the gradient it doesn't matter what the value is - it is only important in the context of the original function. For this reason, the derivative of a constant is defined as 0.

Now that we have found the derivative of each of the terms in our function, we can create a function for the gradient at any given point:

\(y = x^2 + x+2\)

\(\frac{dy}{dx} = 2x +1\)

Therefore to find the gradient at the point where x = 1, substitute this value into our new equation:

\(m = 2(1) + 1 = 3\)

Differentiation from first principles tells us about the concept of differentiation.

Let's consider this curve which is part of a graph that we would like to differentiate. We have chosen two points along it, (x, f (x)) and (x + h, f (x + h)), and we would like to find the gradient at the point (x, f (x)):

We know to find the gradient between these points, we find the change in y divided by the change in x:

\(m = \frac{f(x+h) - f(x)}{x+h-x} = \frac{(x+h)-f(x)}{h}\}\)The closer we move those two points together, the better our estimate of the gradient at (x, f (x)) will be. As h gets closer and closer to 0, the estimate will be better and better. We can write this as the formula:

\(f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)

We know the derivative of x^{2} is 2x, but we can prove this by substituting it into the formula:

\(f(x) x^2\)

\(f'(x) = \lim_{h \rightarrow 0} \frac{(x+h)^2-x^2}{h} = \lim_{h \rightarrow 0} \frac{x^2+2xh+h^2-x^2}{h} = \lim_{h \rightarrow 0}\frac{h(2x+h)}{h} = \lim_{h \rightarrow 0} 2x + h\)

\

Finally, we need to consider what happens at the limit as h approaches 0: h disappears, and we are just left with our answer 2x.

Differentiation can tell us a lot about the nature of graphs and their turning points. These are also known as **critical points** as they are points where the gradient is equal to zero. There are three possibilities when this is the case:

When the graph is quadratic, it's obvious if the critical point is a maximum or minimum, as there is only one, and all you need to do is consider the shape of the graph (using the coefficient of the x^{2} term). However, when there are multiple critical points, it isn't so clear.

In order to determine the nature of a critical point for cubic graphs, you need to check the gradients on either side of it.

Let's consider a local maximum:

We can see that the first part of the graph is **increasing** according to the direction of the graph, then after the critical point, it starts to decrease.

If we found the gradient of the increasing part of the graph, it would be positive, and the decreasing part would be negative. In summary:

\[\frac{dy}{dx} > 0 \quad \text{increasing}\]

\[\frac{dy}{dx} = 0 \quad \text{critical point}\]

\[\frac{dy}{dx} < 0 \quad \text{decreasing}\]

Let's look at determining the nature of a critical point.

\(y = x^2 + 4x +2\)

We already know that the critical point of this graph is going to be a minimum, because the x^{2} has a positive coefficient. However, we'll prove it using differentiation.

First, we need to differentiate the function;

\(y' = 2x + 4\)

Now we need to find the coordinates of the critical point, the x value where the derivative of the function is zero. We can do this by solving the equation \(2x + 4 = 0\), since we know the gradient is zero at that point.

\(2x + 4 = 0 \rightarrow 2x = -4 \rightarrow x = -2\)

Now we can create a simple table and sub in the values of x on either side:

x = -3 | x = -2 | x = -1 |

x' = 2(-3) + 4 = -2 | x' = 0 | x' = 2(-1) + 4 = 2 |

Negative so decreasing | Turning point | Positive so increasing |

Since the gradient on the left is decreasing and the gradient on the right is increasing, we have shown that the turning point is a minimum.

If the gradient on the left would be increasing and the gradient on the right decreasing, the turning point would be a maximum.

Finally, if they are **both increasing or both decreasing**, it must be a stationary point.

A different possibility to determine if a critical point is a maximum, minimum, or stationary point is by using the second derivative, as the second derivative of a graph tells you its curvature.

**A positive curvature**means the graph curves towards the left if considered along the x-axis**(minimum)**.**A negative curvature**means that the graph curves towards the right**(maximum)**.If the second derivative of a function is

**zero**at a certain point, the curvature is zero, and the graph is straight at this point**(stationary point)**.

In our example:

\(y = x^2 + 4x+2\)

\(y' = 2x +4 \)

\(y'' = 2\)

This means that the curvature is positive anywhere on the graph and the critical point is a maximum.

The Product Rule

The Quotient rule

The Chain rule

The product rule can be used to find the derivative of two functions multiplied together. The formula is;

If y = uv, then \(y' = uv' + vu'\)

Where u is the function f(x) and v is the function g(x), and f'(x), g'(x) are their derivatives u' and v'.

Differentiate the function \(y = (x^2 + 1)(x^2+x)\)

We could expand the brackets in this example and find the derivative the usual way, however often using the product rule is faster and less prone to error.

To use the product rule on this function, we need to let \(u = x^2 + 1\) and \(v = x^2 + x\)x.

Next, we need to differentiate them individually:

\(u' = 2x\)

\(v' = 2x+1\)

Finally, we substitute these values into the product formula:

\(y' = (x^2 + 1)(2x+1) + 2x(x^2+x) = 2x^3 + x^2 + 2x + 1 + 2x^3 + 2x^2 = 4x^3 + 3x^2 + 2x +1\)\(y' = \frac{vu' -uv'}{v^2}\)

Where u is the function f (x) and v is the function g (x), and f '(x), g' (x) are their derivatives u' and v'.

Differentiate the function \(y = \frac{x}{x+2}\)

We let u be the numerator, and v be the denominator, ie u = x and v = x + 2, then differentiate them individually as before to get u' = 1 and y' = 1.

Finally, we need to substitute these values into the formula:

\(y' = \frac{(x+2)(1) - (x)(1)}{(x+2)^2}\)

\(y' = \frac{2}{(x+2)^2}\)

The chain rule can be used to find the derivative of a function of a function. The formula is;

\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)Differentiate the function \(y = (x+2)^3\)

We let \(u = x+2\), then substitute this into the main equation such that \(y = u^3\). We then differentiate them both individually, thus finding \(\frac{dy}{du}\) and \(\frac{du}{dx}\);

\(\frac{du}{dx} = u'(x) = 1\)

\(\frac{dy}{du} = y'(u) = 3u^2\)

Finally, we multiply them together to get \(\frac{dy}{du} = 3u^2\) , and substitute u back in to get \(y' = 3(x+2)^2\).

Sometimes we want to differentiate functions where x and y are both in terms of a third variable. In these situations, we need to use parametric differentiation.

\(y = 3t^2 + 2t -3\)

\(x = 4t + 5\)

We can use the chain rule to differentiate in terms of x and y:

\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

We could rearrange the equation involving x to be in terms of t. The above equation could also be written as the following, making it easier to differentiate:

\(\frac{dy}{dx} = \frac{dy}{dt}/\frac{dx}{dt}\)

Let's first try rearranging and multiplying our results: \(x = 4t + 5 \Rightarrow t = \frac{1}{4} (x-5) \rightarrow t' = \frac{1}{4}\)

\(y = 3t^2 + 2t - 3 \rightarrow y' = 6t + 2\)

\(\frac{dy}{dt} \cdot \frac{dt}{dx} = \frac{1}{4} (6t +2) = \frac{3t +1}{2}\)

Now let's try the second method to ensure we get the same answer. All we need to do is differentiate each equation individually with respect to t, and then divide \(\frac{dx}{dt}\) by \(\frac{dy}{dt}\):

\(y = 3t^2 + 2t - 3 \rightarrow y' = 6t + 2\)

\(x = 4t + 5 \rightarrow x'=4\)

\(\frac{dy}{dx} = \frac{6t+2}{4} = \frac{3t+1}{2}\)

We need to use a technique called implicit differentiation to solve this. We can approach each part of the equation separately and write:

\(\frac{d}{dx}x^2 + \frac{d}{dx}y^2 = \frac{d}{dx} 25\)

We know how to differentiate two of the parts. The first stage to differentiating the y part is to differentiate it as normal, but leave \(\frac{dy}{dx}\);

\(2x + \frac{dy}{dx} 2y = 0\)

Now we need to rearrange the equation in terms of \(\frac{dy}{dx}\):

\(\frac{dy}{dx}2y = -2x\)

\(\frac{dy}{dx} = -\frac{2x}{2y} = -\frac{x}{y}\)

Differentiation is a method of finding rates of change, i.e. gradients of functions.

The result of a differentiation calculation is called the

**derivative**of a function.The process of differentiation is represented by \(\frac{dy}{dx}\).

- To differentiate a polynomial:
Decrease the power of x by one

Multiply by the old power

- The derivative of a constant is defined as 0.
- Differentiation from first principles uses the formula, \(f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
- \(\frac{dy}{dx} > 0\) increasing
- \(\frac{dy}{dx} = 0\)critical point
When the derivative is equal to zero, there are three possibilities:

\(\frac{dy}{dx} < 0\) decreasing

The product rule is \(y'= uv'+vu'\)

The quotient rule is \(y' = \frac{vu'-uv'}{v^2}\)

The chain rule is \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

Parametric differentiation uses the formula \(\frac{dy}{dx} = \frac{dy}{dt}/\frac{dx}{dt}\)

Implicit differentiation involves differentiating each part of the equation separately and rearranging for \(\frac{dy}{dx}\)

To differentiate a fraction, you need to use the quotient rule;

y'=(vu'-uv')/v^2

Differentiation is the process of finding a function for the gradient of a given function.

To differentiate a function of a function, you need to use the chain rule; dy/dx=dy/du ⋅ du/dx

What are the three differentiation rules you need to know?

Chain rule, product rule, quotient rule.

When should you use the chain rule?

The chain rule can be used when you are differentiating a composite function.

When should you use the product rule?

This rule can be used when you are differentiating the product of two functions.

When should you use the quotient rule?

This rule is used when you are differentiating a function that is being divided by another function, otherwise known as a quotient function.

What is the product rule?

The product rule is a rule used for differentiation.

When do you use the product rule?

You can use the product rule when you are differentiating the products of two functions.

Already have an account? Log in

Open in App
More about Differentiation

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in

Already have an account? Log in

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up with Email

Already have an account? Log in