Derivative Calculator

This derivative calculator differentiates any function of x and shows f′(x) with every rule named — power, product, quotient, and chain rule — plus a graph of f and f′ and the slope at any point.

Derivative f′(x) —

How to Find a Derivative

Enter a function of x to see its derivative and the rules behind it.

1

Write your function of x

Type f(x) using * or juxtaposition for multiplication (2x means 2·x), and function names like sin, cos, ln, exp, and sqrt — for example x³ + 2x or sin(x)/x.

2

The calculator differentiates term by term

It applies the power, product, quotient, and chain rules, differentiates each function, and simplifies — showing every rule it used so you can follow the working, not just copy the answer.

3

Read f′(x), the graph, and the slope

The simplified derivative appears at the top, f and f′ are plotted together, and an optional point gives the exact slope there. Going the other way? The Quadratic Formula and System of Equations solvers cover algebra.

Frequently Asked Questions

How do you use this derivative calculator?
To use the derivative calculator, type your function of x into the f(x) box — for example x³ + 2x, sin(x)/x, or eˣ. The calculator instantly shows the derivative f′(x) in simplified form and lists every rule it used, from the power rule to the product, quotient, and chain rules. Use * or just juxtaposition for multiplication (2x means 2·x), and function names like sin, cos, tan, ln, log, exp, and sqrt.
Which differentiation rules does it show?
The derivative calculator applies and names the constant rule, power rule, sum and difference rules, product rule, quotient rule, and chain rule, plus the derivatives of common functions (sin, cos, tan, their inverses, ln, log, exp, sqrt, and the hyperbolic functions). For a variable raised to a variable power, such as xˣ, it uses logarithmic differentiation. Each step is labelled with the rule used, so d/dx[x² sin x] is shown as a product rule giving 2x sin x + x² cos x. Repeated application handles nested functions such as sin(3x²), whose derivative is 6x cos(3x²).
Can it evaluate the derivative at a specific point?
Point evaluation is built in. Enter a value in the optional “Evaluate f′ at x =” field and the calculator computes the numeric value of the derivative there, which is the slope of the function at that x. For f(x) = x² the derivative is f′(x) = 2x, so f′(3) = 6. Leave the field blank to see just the symbolic derivative.
What does the graph show?
The graph plots your function f(x) as a solid line and its derivative f′(x) as a dashed line on the same axes, so you can see how the slope of f matches the height of f′. Where f is rising, f′ is positive; where f has a peak or a valley, f′ crosses zero. For f(x) = x² the parabola is paired with the straight line f′(x) = 2x, which crosses zero exactly at the vertex. Changing the function or the range redraws both curves together.

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