Catalogue of composite derivatives

Fifteen worked examples covering all the combinations of elementary functions encountered in Year 5. They form a reference catalogue to review before assessments; each example applies only one idea beyond the previous ones.

Example 1 — Polynomial inside sine: $f(x)=\sin(x^2)$

$f'(x)=\cos(x^2)\cdot 2x$.

Example 2 — Exponential inside cosine: $f(x)=e^{\cos x}$

$f'(x)=e^{\cos x}\cdot(-\sin x) = -\sin x\,e^{\cos x}$.

Example 3 — Logarithm of a polynomial: $f(x)=\ln(1+x^2)$

$f'(x)=\dfrac{1}{1+x^2}\cdot 2x = \dfrac{2x}{1+x^2}$.

Example 4 — Arctangent of an exponential: $f(x)=\arctan(e^x)$

$f'(x)=\dfrac{1}{1+e^{2x}}\cdot e^x = \dfrac{e^x}{1+e^{2x}}$.

Example 5 — Root of a rational function: $f(x)=\sqrt{\dfrac{1+x}{1-x}}$

Set $u=(1+x)/(1-x)$: $u'=\dfrac{(1)(1-x)-(1+x)(-1)}{(1-x)^2}=\dfrac{2}{(1-x)^2}$. Then $f'(x)=\frac{1}{2\sqrt{u}}\cdot u' = \frac{1}{2}\sqrt{\frac{1-x}{1+x}}\cdot\frac{2}{(1-x)^2} = \frac{1}{(1-x)^2}\sqrt{\frac{1-x}{1+x}}.$

Example 6 — Cosine of sine: $f(x)=\cos(\sin x)$

$f'(x)=-\sin(\sin x)\cdot\cos x$.

Example 7 — Exponential of a quadratic: $f(x)=e^{-x^2}$ (the Gaussian)

$f'(x)=e^{-x^2}\cdot(-2x)=-2x\,e^{-x^2}$.

Example 8 — Iterated logarithm: $f(x)=\ln(\ln x)$, $x>1$

$f'(x)=\dfrac{1}{\ln x}\cdot\dfrac{1}{x}=\dfrac{1}{x\ln x}$.

Example 9 — Sine of a logarithm: $f(x)=\sin(\ln x)$

$f'(x)=\cos(\ln x)\cdot\dfrac{1}{x}=\dfrac{\cos(\ln x)}{x}$.

Example 10 — $f(x)=\tan^3(x)$

Set $u=\tan x$: $u'=1/\cos^2 x$. $f'(x)=3u^2\cdot u'=3\tan^2 x/\cos^2 x$.

Example 11 — $f(x)=\arctan(\sqrt{x})$

$f'(x)=\dfrac{1}{1+x}\cdot\dfrac{1}{2\sqrt{x}}=\dfrac{1}{2\sqrt{x}(1+x)}$.

Example 12 — $f(x)=(\sin x)^{\cos x}$ (variable exponent, the "log trick")

The function has both base and exponent depending on $x$: one applies $f=e^{\cos x\,\ln(\sin x)}$. $f'(x) = f(x)\cdot\Bigl[-\sin x\,\ln(\sin x) + \cos x\cdot\frac{\cos x}{\sin x}\Bigr] = (\sin x)^{\cos x}\Bigl[-\sin x\,\ln(\sin x) + \frac{\cos^2 x}{\sin x}\Bigr].$ This is the logarithmic derivative: whenever one sees $f^g$ with both variable, one goes through $\ln f^g = g\ln f$.

Example 13 — $f(x)=x^x$ (a celebrated case)

$\ln f = x\ln x$. Differentiating both sides: $f'/f = \ln x + 1$, hence $f'(x)=x^x(\ln x + 1)$.

Example 14 — Quadratic + root: $f(x)=\sqrt{1+\sin^2 x}$

$f'(x)=\dfrac{1}{2\sqrt{1+\sin^2 x}}\cdot 2\sin x\cos x = \dfrac{\sin x\cos x}{\sqrt{1+\sin^2 x}} = \dfrac{\sin(2x)}{2\sqrt{1+\sin^2 x}}$.

Example 15 — Triple nesting: $f(x)=\ln\bigl(\cos(e^x)\bigr)$

Three circles: $\ln\circ\cos\circ\exp$. $f'(x) = \dfrac{1}{\cos(e^x)}\cdot(-\sin(e^x))\cdot e^x = -e^x\tan(e^x)$.

In brief — Recurring patterns

Pattern	Derivative
$\sin(\varphi)$	$\cos(\varphi)\cdot\varphi'$
$\cos(\varphi)$	$-\sin(\varphi)\cdot\varphi'$
$e^{\varphi}$	$e^{\varphi}\cdot\varphi'$
$\ln(\varphi)$ ($\varphi>0$)	$\varphi'/\varphi$
$\varphi^n$ ($n$ fixed)	$n\varphi^{n-1}\cdot\varphi'$
$\sqrt{\varphi}$ ($\varphi>0$)	$\varphi'/(2\sqrt{\varphi})$
$\arctan(\varphi)$	$\varphi'/(1+\varphi^2)$
$f^g$ (both variable)	go through $e^{g\ln f}$ (logarithmic derivative)

Links

Topics: Derivatives
Concepts: Composite derivative · Logarithmic derivative · Chain rule
Skills: Differentiating · Using formulae