Derivative Calculator with Steps

Function f(x)

Use ^ for powers and * for multiplication. Supports sin, cos, tan, exp, ln, log, sqrt, e, pi.

Derivative order

Evaluate at x = … (optional)

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Step-by-step solution

Worked example: f(x) = x³ + 2x² − 5x + 1

Step 1 — Sum rule: differentiate term by term.

Step 2 — Power rule on x³ → 3x². Constant multiple + power rule on 2x² → 4x. Power rule on −5x → −5. Constant rule on 1 → 0.

Answer: f′(x) = 3x² + 4x − 5.

Evaluate at x = 2: f′(2) = 3(4) + 4(2) − 5 = 12 + 8 − 5 = 15. And f(2) = 8 + 8 − 10 + 1 = 7, so the tangent line at x = 2 is y = 15(x − 2) + 7, i.e. y = 15x − 23.

Type this exact function into the calculator above with x = 2 in the evaluate box — you'll get the same steps, the same 15, and the same tangent line.

The 5 differentiation rules (with one-line examples)

Rule	When to use it	Example
Power rule	x raised to a number	d/dx[x⁵] = 5x⁴
Constant multiple	a number times a function	d/dx[3x²] = 6x
Sum / difference	terms added or subtracted	d/dx[x² + x] = 2x + 1
Product rule	two functions multiplied	d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x)
Quotient rule	one function divided by another	d/dx[(x²+1)/(x−3)] — see steps above
Chain rule	a function inside a function	d/dx[sin(x²)] = 2x·cos(x²)

The chain rule, properly explained

The chain rule is where most marks are lost, so it deserves its own section. Think outside-in:

Differentiate the outer function, leaving the inner part untouched.
Multiply by the derivative of the inner function.

For sin(x²): the outer derivative is cos(x²); the inner derivative of x² is 2x. Multiply: 2x·cos(x²). The classic mistake is stopping at cos(x²) and forgetting the × 2x. The calculator's steps call this out explicitly every time it applies the chain rule.

Second derivatives

The first derivative f′(x) measures the rate of change; the second derivative f″(x) measures how that rate itself changes — the curvature of the graph. Select “2nd derivative” above and the calculator differentiates twice, showing both rounds of steps. For f(x) = x⁴: f′(x) = 4x³, then f″(x) = 12x².

Why the steps are shown, not hidden

Most online derivative tools either hide the working behind an “expand” button, require an account, or route your homework through an AI and make you wait. This calculator does the differentiation symbolically in your browser — no account, no waiting — and prints every rule it applies, in order, with the intermediate expression. If a function uses a form the step engine doesn't cover yet (like xˣ), it still gives you the exact symbolic answer and says so honestly instead of inventing steps.

Common mistakes to avoid

Forgetting the chain rule's inner derivative — d/dx[sin(3x)] is 3cos(3x), not cos(3x).
Sign errors on trig functions — d/dx[cos(x)] = −sin(x). The minus sign is the whole battle.
Product rule ≠ multiply the derivatives — (uv)′ is u′v + uv′, not u′v′.
Quotient rule order — it's (u′v − uv′)/v². Swap the subtraction and the sign flips.
Constants vanish — the +1 in x³+2x²−5x+1 contributes nothing to the derivative.

Frequently asked questions

How do you find a derivative step by step?

Identify the structure of the function, then apply one rule at a time: constants become 0, xⁿ becomes n·xⁿ⁻¹ (power rule), sums split term by term, products use the product rule, fractions use the quotient rule, and nested functions use the chain rule. This calculator names the rule it applies at every step so you can follow the logic.

What is the power rule?

The power rule says d/dx[xⁿ] = n·xⁿ⁻¹: bring the exponent down as a multiplier, then subtract one from the exponent. For example, d/dx[x⁵] = 5x⁴, and d/dx[x³] = 3x². It handles every polynomial term.

When do I use the product rule vs the quotient rule?

Use the product rule when two functions are multiplied, like x²·sin(x): (uv)′ = u′v + uv′. Use the quotient rule when one function is divided by another, like (x²+1)/(x−3): (u/v)′ = (u′v − uv′)/v². Multiplication → product rule; division → quotient rule.

What is the chain rule in simple terms?

The chain rule handles functions inside functions: differentiate the outside, leave the inside alone, then multiply by the derivative of the inside. For sin(x²): the outside derivative is cos(x²), the inside derivative is 2x, so the answer is 2x·cos(x²). Forgetting to multiply by the inside derivative is the most common student mistake.

What is a second derivative?

The second derivative is the derivative of the derivative — it measures how the rate of change itself is changing (curvature/concavity). For f(x) = x⁴: f′(x) = 4x³, then f″(x) = 12x². Select “2nd derivative” in the calculator above to see both rounds of steps.

Can this calculator do partial derivatives?

Not yet — this calculator differentiates single-variable functions of x only. If you type another symbol it will tell you so rather than guessing. Partial derivatives for multivariable functions are on the roadmap.

How do I type e^x and ln(x) into the calculator?

Type e^(3*x) for exponentials (the ^ and * matter), and ln(x) for the natural log — the calculator converts it automatically. Use sqrt(x) for square roots, sin(x), cos(x), tan(x) for trig, and ^ for any power, e.g. x^3.