DerivativesMedium

Chain rule — composite function

Calculate the derivative of f(x) = sin(ln(x² + 1)) using the chain rule.

Given data

f(x) = sin(ln(x² + 1))

Steps

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Identify the layered structure: f = sin(u), u = ln(v), v = x² + 1.
Write your working — the numerical result must appear in the text (logica)
Differentiate the outer layer: d/du sin(u) = ?
Differentiate the middle layer: d/dv ln(v) = ?
Differentiate the inner layer: d/dx (x² + 1) = ?
Multiply everything: f'(x) = cos(ln(x²+1)) · 1/(x²+1) · 2x = ?

Full worked solution

Identify the layered structure: f = sin(u), u = ln(v), v = x² + 1.
$f = \sin(u),\quad u = \ln(v),\quad v = x^2 + 1$
$f(x) = \sin(\ln(x^2+1))$ has three layers: outer $\sin$ , middle $\ln$ , inner $x^2+1$ . We differentiate from outside in using the chain rule.
Differentiate the outer layer: d/du sin(u) = ?
$\dfrac{d}{du}\sin(u) = \cos(u)$
$\frac{d}{du}\sin(u) = \cos(u) = \cos(\ln(x^2+1))$ . The derivative of the outer sine function evaluated at the middle layer.
Differentiate the middle layer: d/dv ln(v) = ?
$\dfrac{d}{dv}\ln(v) = \dfrac{1}{v}$
$\frac{d}{dv}\ln(v) = \frac{1}{v} = \frac{1}{x^2+1}$ . The derivative of the natural logarithm of the inner function.
Differentiate the inner layer: d/dx (x² + 1) = ?
$\dfrac{d}{dx}(x^2 + 1) = 2x$
$\frac{d}{dx}(x^2+1) = 2x$ . The derivative of the innermost polynomial.
Multiply everything: f'(x) = cos(ln(x²+1)) · 1/(x²+1) · 2x = ?
$f'(x) = \dfrac{2x \cos(\ln(x^2+1))}{x^2+1}$
$f'(x) = \cos(\ln(x^2+1)) \cdot \frac{1}{x^2+1} \cdot 2x = \mathbf{\frac{2x\,\cos(\ln(x^2+1))}{x^2+1}}$ . Multiplying the derivatives of all three layers gives the final result.

Result:f'(x) = 2x·cos(ln(x²+1)) / (x²+1).

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