havo B vwo A and B
Derivative calculator
Quick answer
Power rule: the derivative of a·xⁿ is n·a·xⁿ⁻¹. A constant becomes 0.
f(x) = 3x^2 − 5x + 4 gives f'(x) = 6x − 5. The slope at x = 2 is f'(2) = 7.
Calculator
Type a function built from powers of x, such as 3x^2 - 5x + 4, 2/x or x^(1/2). This calculator uses the power rule and the sum rule. It cannot do the product, quotient or chain rule yet, and it does not handle functions such as eˣ, ln x or sin x. Decimals appear with a comma (0,5 rather than 0.5), as in Dutch schools.
What is a derivative?
The derivative f'(x) tells you, for every x, the slope of the graph of f: how steeply it goes up or down there. If f'(a) is positive the graph is rising at x = a, if it is negative the graph is falling, and if f'(a) = 0 the graph is flat for a moment, for example at a vertex. Dutch books call the derivative de afgeleide and the slope de helling.
For power functions you use the power rule: the derivative of a·xⁿ is n·a·xⁿ⁻¹. The exponent moves to the front of the term and goes down by 1. With the sum rule you differentiate each term of a sum on its own.
A fraction or a root has to become a power first: 2/x = 2x^-1 and √x = x^(1/2). The derivative of 2x^-1 is -2x^-2, and the derivative of x^(1/2) is 0,5x^(-1/2).
The derivative also gives you the tangent line at a point: its slope is f'(a), and the line passes through the point (a, f(a)).
Examples
Given f(x) = 2x^3 − 6x. Find f'(x) and the slope at x = 2.
- First term2x^3 → 3 · 2 · x^2 = 6x^2
- Second term-6x → 1 · (-6) · x^0 = -6
- Derivativef'(x) = 6x^2 − 6
- Slopef'(2) = 18
Differentiate f(x) = 2/x.
- As a powerf(x) = 2x^-1
- Power rule2x^-1 → -1 · 2 · x^-2 = -2x^-2
So f'(x) = -2x^-2, which is −2 divided by x².
Find an equation of the tangent line to f(x) = x^2 − 4x + 1 at the point where x = 3.
- Derivativef'(x) = 2x − 4
- Slopef'(3) = 2
- Pointf(3) = -2
- Find bb = -2 − 2 · 3 = -8
The tangent line is y = 2x − 8.
Common mistakes
Wrong
Lowering the exponent by 1 but not putting it in front.
Right
Do both: the derivative of 2x^3 is 3 · 2 · x^2 = 6x^2.
Wrong
Keeping the constant in the derivative.
Right
A constant has derivative 0: for x^2 − 4x + 1 you get f'(x) = 2x − 4.
Wrong
For the slope at x = 3, working out the function value f(3).
Right
The slope is f'(3), not f(3). For the tangent line you need both.
Wrong
With x^-1, taking the exponent “one lower” to 0.
Right
-1 - 1 = -2, so the derivative of x^-1 is -x^-2.
To practice with an explanation for every mistake, try derivatives practice.
Derivatives on the Dutch exam
The 2027 central exam syllabus for wiskunde B havo (subdomain D3) asks you to find the derivative of power functions with rational exponents and to use the sum and difference rules. The chain rule belongs there too, and this calculator cannot do it yet.
In wiskunde A vwo the derivative sits in subdomain D3, for the standard functions except the sine. In wiskunde B vwo you use the derivative in subdomain C1, among other things to set up the equation of a tangent line, and subdomain C2 covers all the differentiation rules.
Next step
- How does this work? The explanation and examples are further up this page.
- Test yourself Derivatives practice, with an explanation after every answer.
Learning path
- Related: Quadratic formula calculator
Frequently asked questions
What is the power rule?
The derivative of a·xⁿ is n·a·xⁿ⁻¹: the exponent moves to the front and goes down by 1. A constant on its own has derivative 0.
What does the derivative at a point tell you?
f'(a) is the slope of the graph of f at the point where x = a. For f(x) = x^2 − 4x + 1 the derivative is f'(x) = 2x − 4, so at x = 2 the slope is 0: that is where the vertex is.
How do you write 2/x or √x as a power?
2/x = 2x^-1 and √x = x^(1/2). After that the power rule works as usual: the derivative of 2x^-1 is -2x^-2.