Derivative Calculator guide
Derivative Calculator: Differentiate Step by Step
Use our derivative calculator to find first, second and third derivatives, the gradient at a point and the tangent line, with the differentiation rules explained.
By the AllCalculator editorial teamUpdated 15 October 20262 min read
A derivative tells you how fast something is changing: the gradient of a curve at any point. Our Derivative Calculator finds the derivative of a function, simplifies it and works out the gradient and tangent line at any point you choose.
How to use the calculator (Derivative Calculator)
- Type your function of x, for example x^3 × cos(x).
- Choose the first, second or third derivative.
- Optionally, enter a value of x to see the gradient there and the equation of the tangent line.
Other letters are treated as constants, so the calculator gives partial derivatives with respect to x.
The main rules
| Rule | Function | Derivative |
|---|---|---|
| Power | xⁿ | n·xⁿ⁻¹ |
| Constant multiple | k·f(x) | k·f′(x) |
| Sum | f + g | f′ + g′ |
| Product | f·g | f′g + fg′ |
| Quotient | f / g | (f′g − fg′) / g² |
| Chain | f(g(x)) | f′(g(x))·g′(x) |
And the standard derivatives: sin x → cos x, cos x → −sin x, tan x → sec² x, eˣ → eˣ, ln x → 1/x.
A worked example
With the calculator's default settings, f(x) = x³ cos(x). Using the product rule:
- f′(x) = 3x² cos(x) − x³ sin(x)
- At x = 1: f′(1) = 3cos(1) − sin(1) ≈ 0.779
So the curve is rising at x = 1, and the tangent line there is about y = 0.779x − 0.239.
Uses of derivatives
- Stationary points: where f′(x) = 0, the curve has a maximum, minimum or point of inflection.
- Second derivative test: if f″(x) > 0 it's a minimum, and if f″(x) < 0 it's a maximum.
- Rates of change: speed is the derivative of distance, and acceleration is the derivative of speed.
See the curve with our Graphing Calculator, or go the other way with our Integral Calculator.
Frequently asked questions
Does the calculator show the steps?
It shows each derivative in turn, simplified, and the value at your chosen point. Use the rules above to follow the working.
Can it do partial derivatives?
Yes. Any letter other than x is treated as a constant, so you get the partial derivative with respect to x.
Why does the answer look different from my textbook?
There's often more than one correct way to write a derivative. Check by putting the same value of x into both.
Differentiation gets easier with practice. Use the Derivative Calculator to check your answers as you work.
Sources and further reading
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.