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Calculus basics

Chain rule

Differentiates a function of a function: outside derivative times inside derivative.

dydx=dydu·dudx
Bracket formddx(ax+b)n=na(ax+b)n−1
Function notation(f(g(x)))′=f′(g(x))g′(x)

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−2.5−2−1.5−1−0.50.511.522.5−5050100150
  • At x = 1: f(x) = 27, slope f′(x) = 54 (coral tangent). Dashed: the derivative f′.

What the symbols mean

D
the value of dy/dx at the chosen x
a
in the calculator, the coefficient of x inside (ax+b)n
b
the constant inside the bracket
n
the outer power
x
the point
u
the inside function
y
the whole function, written in terms of u

When to use it

Use it whenever something more complicated than x sits inside a power, root, trig, exponential or log: (3x−1)5, x2+1, sin⁡(2x), ex2. Identify the inside u, differentiate the outside as if u were x, then multiply by du/dx.

Worked example

Differentiate y = (2x + 1)³ and find the slope at x = 1 (a = 2, b = 1, n = 3).

  1. Chain rule: derivative of the outside (leaving the inside alone) times derivative of the inside.

    dydx=dydu·dudx

  2. Inside: u=2⁢x+1, so dudx=2. Outside: y=u3, so dydu=3u2.

  3. Evaluate the inside at x:

    u=a⁢x+b
    =2·1+1
    =2+1
    =3

  4. Derivative of the outside at that u:

    dydu=n⁢un−1
    =3·33−1
    =3·32
    =3·9
    =27

  5. Multiply by the derivative of the inside:

    D=dydu⁢a
    =27·2
    =54

Answer dy/dx = 54 at x = 1

Why it works

A small change in x changes u by about dudx times as much, and that change in u changes y by about dydu times as much. Rates of change multiply along the chain.

Common mistakes

  • Forgetting to multiply by the derivative of the inside: (2x+1)3 differentiates to 6(2x+1)2, not 3(2x+1)2.
  • Changing the inside when differentiating the outside: keep (2x+1) as it is.
  • Expanding big brackets instead; the chain rule is far quicker and less error-prone.