Differentiation calculator with steps
Enter a function of x and this tool differentiates it, simplifies the result, and gives you the second derivative too. Supply a value of x and it also returns the gradient at that point, the coordinates on the curve, and the equation of the tangent — the standard follow-up parts in A Level differentiation questions.
Enter a function of x
dy/dx = 6 * x - 5
Second derivative: d²y/dx² = 6
Gradient at x = 2: 7
Point on the curve: (2, 4)
Tangent: y = 7x − 10
Working
- Differentiate term by term with respect to x.
- dy/dx = 6 * x - 5
- d²y/dx² = 6
- At x = 2: gradient = 7 and y = 4.
- Tangent: y − 4 = 7(x − 2) → y = 7x − 10
The method behind it
- Power rule: d/dx (axⁿ) = anxⁿ⁻¹.
- Chain rule for a function inside a function; product and quotient rules for combinations.
- Gradient at a point = value of dy/dx there; tangent: y − y₁ = m(x − x₁).
How to use this tool
1. Type your function
Use ^ for powers and * for multiplication, for example 3x^2 - 5x + 2, sin(x), e^x or ln(x).
2. Read the derivative
The tool shows dy/dx in simplified form, plus the second derivative d²y/dx² for turning-point questions.
3. Evaluate at a point
Enter a value of x to get the gradient there, the point on the curve, and the tangent equation.
Frequently asked questions
How do you differentiate x to a power?
Multiply by the power, then reduce the power by one: the derivative of axⁿ is anxⁿ⁻¹. So the derivative of 3x⁴ is 12x³.
What does the second derivative tell you?
At a stationary point, a positive second derivative means a minimum, a negative value means a maximum, and zero means you must test further — it may be a point of inflection.
How do you find the equation of a tangent?
Find the gradient m by substituting x into dy/dx, find the y coordinate on the curve, then use y − y₁ = m(x − x₁).
Which functions can this differentiate?
Polynomials, roots and negative powers, trigonometric functions, exponentials and logarithms, and products, quotients and compositions of them.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.