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Derivatives with rule-named steps — and a slope check

Type a function. Each step names the rule it applies — Power rule, Chain rule, Product rule — and then the answer is tested against the actual slope of your function at real points. A derivative that matches the slope is a derivative you can trust.

  • ✓100% free — no paywall, ever
  • ✓No sign-up, no app
  • ✓Every answer verified before you see it

In short

A derivative calculator finds the rate at which a function changes. VerifySolve differentiates polynomials, trig, exponential and logarithmic functions using the power, product, quotient and chain rules, naming the rule used at every step. Each derivative is then verified numerically: the slope it predicts is compared with the function’s actual slope at a test point.

The derivative rules

Six rules generate almost every derivative in a first calculus course. The solver names them as it goes, so the steps double as a study guide.

Power rule

ddx xn=nxn−1\dfrac{d}{dx}\, x^n = n x^{n-1}

Product rule

ddx [f⋅g]=f′g+fg′\dfrac{d}{dx}\,[f \cdot g] = f'g + fg'

Quotient rule

ddx ⁣[fg]=f′g−fg′g2\dfrac{d}{dx}\!\left[\dfrac{f}{g}\right] = \dfrac{f'g - fg'}{g^2}

Chain rule

ddx f(g(x))=f′(g(x))⋅g′(x)\dfrac{d}{dx}\, f(g(x)) = f'(g(x)) \cdot g'(x)

Trig & exponential

ddxsin⁡x=cos⁡xddxex=ex\dfrac{d}{dx} \sin x = \cos x \qquad \dfrac{d}{dx} e^x = e^x

Logarithm

ddxln⁡x=1x\dfrac{d}{dx} \ln x = \dfrac{1}{x}

Worked examples — slope-checked

Every derivative below was verified the same way the live solver verifies yours: formula value vs. measured slope, at several points.

f(x) = 3x² + 2x − 5

  1. 1.Power rule on each term: 3x² → 6x, 2x → 2
  2. 2.Constant rule: −5 → 0

f′(x) = 6x + 2

Slope check at x = 1: formula gives 8; the function’s measured slope is 8 ✓.

f(x) = sin(3x)

  1. 1.Chain rule: derivative of the outer sin is cos, keeping the inside: cos(3x)
  2. 2.Multiply by the derivative of the inside, 3

f′(x) = 3cos(3x)

Verified numerically at four sample points ✓. Forgetting the ×3 is the classic chain-rule miss — the slope check would catch it.

f(x) = (3x + 1)²

  1. 1.Chain rule: 2(3x + 1) · 3
  2. 2.Simplify: 6(3x + 1)

f′(x) = 18x + 6

Verified against the numeric slope at several points ✓.

f(x) = e^(2x)

  1. 1.Chain rule: e^(2x) stays, times the derivative of 2x

f′(x) = 2e^(2x)

Verified numerically ✓ — exponentials keep their shape; only the scale changes.

f(x) = ln(x)

  1. 1.Logarithm rule: the derivative of ln x is 1/x

f′(x) = 1/x

Verified against the measured slope at positive sample points ✓.

Common mistakes — and how to dodge them

Dropping the inside derivative (chain rule)

The derivative of sin(3x) is 3cos(3x), not cos(3x). The chain rule’s whole point is the extra factor from the inside function. If your answer’s slope check fails by exactly that factor, you know what you dropped.

Treating a product like a sum

The derivative of x·sin(x) is not 1·cos(x). Products need the product rule: each factor takes a turn being differentiated while the other stays put. Same warning for quotients — there is a rule, use it.

Applying the power rule to eˣ or sin x

The power rule is only for x raised to a number. eˣ has the variable in the exponent — its derivative is itself (times the inside derivative). Pattern-match the shape before choosing the rule.

Derivative Calculator — questions

How is a derivative answer verified?

Numerically, the honest way: the calculator picks real x-values, measures the slope of your original function there with a tiny secant (rise over run as the run shrinks), and compares it with the derivative formula’s value at the same points. If they agree at several points, the derivative is right — the check is shown under every answer.

Which derivative rules are covered?

The power rule, constant and sum/difference rules, the product rule, the quotient rule, and the chain rule — applied to polynomials and to sin, cos, exp, ln and square roots, including combinations like sin(3x), (3x + 1)² and x·eˣ. Each step names the rule it used.

What is not covered yet?

Inverse trig functions (arctan and friends), hyperbolic functions, and special cases like x^x that need logarithmic differentiation. Type one and the calculator says so plainly instead of returning a confident wrong answer.

How do I type my function?

Just the expression: 3x^2 + 2x - 5, sin(3x), (3x+1)^2, exp(2x), ln(x). You can also write f(x) = … first — the calculator strips the label. Use ^ for powers and * for multiplication (or just write 3x).

What is the difference between f′(x) and dy/dx?

Notation only — both mean “the derivative”. f′(x) is common when the function is named f; dy/dx is common when y is written as an expression in x. For equations where y cannot be isolated, use our implicit differentiation calculator.

Is the derivative calculator free?

Yes — completely free, no sign-up, no app, and the steps and verification are never paywalled.

Keep solving

Integral Calculator →

Derivatives in reverse — antiderivatives, verified by differentiating the answer.

Implicit Differentiation Calculator →

dy/dx for relations like x² + y² = 25 that you cannot solve for y.

Exponent Calculator →

The power fluency that makes the power rule automatic.

Algebra Calculator →

Equations, systems and factoring — verified the same way.

A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.