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Reference

Differentiation rules.

The rules the engine applies step by step.

Constant rule

rule
d/dx c = 0

The derivative of a constant is zero.

Variable rule

rule
d/dx x = 1

The derivative of a variable with respect to itself is 1.

Power rule

rule
d/dx (u^n) = n·u^(n−1)·u′

Bring the exponent down, subtract one, multiply by the inner derivative.

Sum rule

rule
d/dx (u + v) = u′ + v′

The derivative of a sum is the sum of the derivatives.

Product rule

rule
d/dx (u·v) = u′v + uv′

Derivative times the other, plus the other times the derivative.

Quotient rule

rule
d/dx (u/v) = (u′v − uv′)/v²

Low-high over the square of the bottom.

Chain rule

rule
d/dx f(g(x)) = f′(g(x))·g′(x)

Multiply the outer derivative by the inner derivative.

Exponential rule

rule
d/dx (e^u) = e^u·u′

The derivative of e raised to anything is itself, times the inner derivative.

Log rule

rule
d/dx (ln u) = u′/u

The derivative of a natural log is one over the inside, times the inner derivative.

Log base-a rule

rule
d/dx log_a(u) = u′ / (u·ln a)

The derivative of a logarithm with an arbitrary base.

Trig: sin / cos

rule
d/dx sin u = cos u · u′; d/dx cos u = −sin u · u′

The derivatives of sine and cosine, with the chain rule applied.

Trig: tan

rule
d/dx tan u = sec²(u) · u′

The derivative of tangent is secant squared.

Trig: sec

rule
d/dx sec u = sec u · tan u · u′

Derivative of secant.

Trig: csc

rule
d/dx csc u = −csc u · cot u · u′

Derivative of cosecant.

Trig: cot

rule
d/dx cot u = −csc² u · u′

Derivative of cotangent.

Inverse trig: asin

rule
d/dx asin u = u′ / sqrt(1−u²)

Derivative of arcsine.

Inverse trig: acos

rule
d/dx acos u = −u′ / sqrt(1−u²)

Derivative of arccosine.

Inverse trig: atan

rule
d/dx atan u = u′ / (1 + u²)

The most common inverse trig derivative.

Inverse trig: asec

rule
d/dx asec u = u′ / (|u|·sqrt(u²−1))

Derivative of arcsecant.

Inverse trig: acsc

rule
d/dx acsc u = −u′ / (|u|·sqrt(u²−1))

Derivative of arccosecant.

Inverse trig: acot

rule
d/dx acot u = −u′ / (1 + u²)

Derivative of arccotangent.

Square root

rule
d/dx sqrt(u) = u′ / (2·sqrt(u))

Half over the square root, times the inner derivative.

Hyperbolic: sinh

rule
d/dx sinh u = cosh u · u′

Derivative of hyperbolic sine.

Hyperbolic: cosh

rule
d/dx cosh u = sinh u · u′

Derivative of hyperbolic cosine.

Hyperbolic: tanh

rule
d/dx tanh u = (1−tanh²u) · u′

Derivative of hyperbolic tangent.

Hyperbolic: sech

rule
d/dx sech u = −sech u · tanh u · u′

Derivative of hyperbolic secant.

Hyperbolic: csch

rule
d/dx csch u = −csch u · coth u · u′

Derivative of hyperbolic cosecant.

Hyperbolic: coth

rule
d/dx coth u = −csch²u · u′

Derivative of hyperbolic cotangent.

Inv. hyperbolic: asinh

rule
d/dx asinh u = u′ / sqrt(u²+1)

Derivative of inverse hyperbolic sine.

Inv. hyperbolic: acosh

rule
d/dx acosh u = u′ / sqrt(u²−1)

Derivative of inverse hyperbolic cosine.

Inv. hyperbolic: atanh

rule
d/dx atanh u = u′ / (1−u²)

Derivative of inverse hyperbolic tangent.

Inv. hyperbolic: asech

rule
d/dx asech u = −u′ / (u·sqrt(1−u²))

Derivative of inverse hyperbolic secant.

Inv. hyperbolic: acsch

rule
d/dx acsch u = −u′ / (|u|·sqrt(u²+1))

Derivative of inverse hyperbolic cosecant.

Inv. hyperbolic: acoth

rule
d/dx acoth u = u′ / (1−u²)

Derivative of inverse hyperbolic cotangent.

Signum / sign

rule
d/dx sign u = 0 (u ≠ 0)

sign(u) is non-differentiable at u=0; treated as 0 elsewhere.

A worked example

Differentiate y = sin(x²) with respect to x.

  1. 1
    Recognise a composition
    u = x², f(u) = sin u
  2. 2
    Differentiate the outer function
    d/dx sin u = cos u · u′
  3. 3
    Differentiate the inner function (power rule)
    d/dx x² = 2x
  4. 4
    Substitute back
    y′ = cos(x²) · 2x = 2x·cos(x²)
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