Derivative Calculator

Differentiate any function and see which rule is applied at every step.

Try an example:

What a derivative measures

The derivative $f^{'} (x)$ gives the instantaneous rate of change of $f$ at $x$ — the slope of the tangent line to the graph at that point. It's defined as a limit:

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

In practice you rarely use the limit directly. A handful of rules cover almost every function you'll meet.

How to use the derivative calculator

Type or paste your problem into the box above — for example d/dx 4x^3 - 2x + 7. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

The core rules

Power rule: $\frac{d}{d x} x^{n} = n x^{n - 1}$
Product rule: $(uv)^{'} = u^{'} v + u v^{'}$
Quotient rule: $(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
Chain rule: $\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)$
Standard results: $(sin x)^{'} = cos x$ , $(cos x)^{'} = - sin x$ , $(e^{x})^{'} = e^{x}$ , $(ln x)^{'} = \frac{1}{x}$

How to approach a derivative

Identify the outermost structure: a sum, a product, a quotient or a composition.
Apply the matching rule, then differentiate the smaller pieces it leaves you with.
Simplify. Factoring out common terms makes the result easier to use — for example when solving $f^{'} (x) = 0$ .

Worked example: $\frac{d}{d x} [x^{2} sin x]$

Spot the product $u = x^{2}$ and $v = sin x$ .
Differentiate each part $u^{'} = 2 x$ and $v^{'} = cos x$ .
Apply the product rule $u^{'} v + u v^{'} = 2 x sin x + x^{2} cos x$
Factor $x (2 sin x + x cos x)$

Common mistakes to avoid

Multiplying derivatives in a product

$(uv)^{'}$ is not $u^{'} v^{'}$ . For $x^{2} sin x$ that would wrongly give $2 x cos x$ .

Forgetting the chain rule

$\frac{d}{d x} (3 x + 1)^{5} = 5 (3 x + 1)^{4} \cdot 3$ . The inner derivative, 3, is easy to drop.

Wrong order in the quotient rule

The numerator is $u^{'} v - u v^{'}$ , in that order. Swapping the terms flips the sign of the answer.

Related calculators

Browse all calculators

Questions students ask

Does it show which rule is used?

Yes. Each step names the rule being applied and the pieces it's applied to, so you can follow the reasoning, not just the answer.

Can it find second derivatives?

Yes. Ask for the second derivative of x^4 - 3x^2, or differentiate the previous answer with a follow-up.

Can it do implicit differentiation?

Yes, for relations like $x^{2} + y^{2} = 25$ . It differentiates both sides with respect to $x$ and solves for $\frac{d y}{d x} = - \frac{x}{y}$ .

Can it evaluate the derivative at a point?

Yes. Ask for the slope at $x = 2$ , or for the equation of the tangent line there.