Derivative Calculator — Step by Step

Differentiate any function and see which rule is used at every step — power, product, quotient and chain rules, plus trig, exponential and log derivatives. Get the 2nd and 3rd derivatives, the slope and tangent line at a point, and a graph.

Use ^ for powers, sqrt(), sin(), cos(), tan(), ln(), log() (base 10), e^x, atan(), |x|.

Try:

Evaluate at x =

Derivative

f′(x) = 2x sin(3x) + 3x² cos(3x)

Step-by-step differentiation

First derivative

Product rule
(uv)′ = u′v + uv′, u = x², v = sin(3x)ddx[x² sin(3x)] = 2x sin(3x) + x² · 3 cos(3x) = 2x sin(3x) + 3x² cos(3x)
- Power rule
  ddx[xⁿ] = nxⁿ⁻¹ddx[x²] = 2x^{2 − 1} = 2x
- Chain rule (sin)
  ddx[f(u)] = f′(u)·u′, u = 3xddx[sin(3x)] = cos(3x) · 3 = 3 cos(3x)

Graph of f and f′

x-range:

f(x)f′(x)

Where f′ is above the axis, f is increasing; where f′ crosses zero, f has a peak, a dip or a flat point.

How to use the steps

Each line names the rule, shows it applied, then the simplified result. Indented lines work out the inner pieces (u′ and v′) the outer rule needs.
Results are computed symbolically and exactly — not estimated — so you can use them to check homework.
Natural log is ln(x); log(x) means base 10. Angles in trig functions are in radians.

Derivative formulas

Definitionf′(x) = lim_h→0 f(x + h) − f(x)h

Power ruled/dx [xⁿ] = n xⁿ⁻¹

Constant multiple & sumd/dx [c·f] = c·f′, d/dx [f ± g] = f′ ± g′

Product rule(uv)′ = u′v + uv′

Quotient rule(uv)′ = u′v − uv′v²

Chain ruled/dx [f(g(x))] = f′(g(x)) · g′(x)

Exponentialsd/dx [e^x] = e^x, d/dx [a^x] = a^x ln a

Logarithmsd/dx [ln x] = 1x, d/dx [log_a x] = 1x ln a

Trigsin → cos, cos → −sin, tan → sec²

More trigsec → sec tan, csc → −csc cot, cot → −csc²

Inverse trigsin⁻¹ x → 1√1 − x², tan⁻¹ x → 11 + x²

Tangent line at x = ay = f(a) + f′(a)(x − a)

Worked examples

Differentiate f(x) = 3x⁴ − 5x² + 2x − 7.
1. Power rule on each term: 3·4x³ − 5·2x + 2 − 0
Answer: f′(x) = 12x³ − 10x + 2
Product rule: f(x) = x² eˣ.
1. u = x², v = e^x → u′ = 2x, v′ = e^x
2. f′ = 2x·e^x + x²·e^x
Answer: f′(x) = x eˣ (x + 2)
Quotient rule: f(x) = (x + 1)/(x − 1).
1. f′ = 1·(x − 1) − (x + 1)·1(x − 1)²
Answer: f′(x) = −2/(x − 1)²
Chain rule: f(x) = (3x² + 1)⁴.
1. Outside: 4(3x² + 1)³, inside derivative: 6x
2. f′ = 4(3x² + 1)³ · 6x
Answer: f′(x) = 24x(3x² + 1)³
Tangent line to f(x) = x³ at x = 2.
1. f(2) = 8, f′(x) = 3x² → f′(2) = 12
2. y = 8 + 12(x − 2)
Answer: y = 12x − 16

Tips & common mistakes

Rewrite roots and fractions as powers first: √x = x^1/2, 1/x² = x⁻² — then the power rule does the work.
Chain rule in words: derivative of the outside (leave the inside alone) × derivative of the inside.
(uv)′ is not u′v′ — forgetting the product rule is the most common mistake.