The product, quotient and chain rules of differentiation, and the derivatives of standard functions $x^n$ for $n \in Q$, $e^x$, $\ln(x)$, $\sin(x)$, $\cos(x)$ and $\tan(x)$

What this dot point is asking

VCAA wants fluent by-hand differentiation of any function built from the standard library (polynomials, $e^x$, $\ln x$, $\sin x$, $\cos x$, $\tan x$) using the four standard rules. Paper 1 almost always opens with a differentiation question and rewards clean, factored answers.

Standard derivatives

Memorise these. They appear in nearly every paper.

The power rule $\frac{d}{dx}(x^n) = n x^{n-1}$ extends to all rational $n$: e.g. $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2} x^{-1/2}$, and $\frac{d}{dx}(x^{-2}) = -2 x^{-3}$.

The derivative of $\ln(x)$ is $\frac{1}{x}$ for $x > 0$. For the extended form, $\frac{d}{dx}(\ln|x|) = \frac{1}{x}$ for $x \neq 0$.

The sum rule

You differentiate term by term. Constants come out: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$.

The product rule

If $y = u(x) v(x)$, then

In words: derivative of the first times the second, plus the first times derivative of the second.

Example. Differentiate $y = x^2 e^x$.

$u = x^2$, $v = e^x$. $u' = 2 x$, $v' = e^x$.

$\frac{dy}{dx} = 2 x \cdot e^x + x^2 \cdot e^x = e^x (2 x + x^2) = x e^x (2 + x)$.

The quotient rule

If $y = \frac{u(x)}{v(x)}$, then

Note the sign: $u' v$ minus $u v'$ in that order.

Example. Differentiate $y = \frac{x}{\ln(x)}$ for $x > 0$, $x \neq 1$.

$u = x$, $v = \ln(x)$. $u' = 1$, $v' = \frac{1}{x}$.

$\frac{dy}{dx} = \frac{1 \cdot \ln(x) - x \cdot \frac{1}{x}}{[\ln(x)]^2} = \frac{\ln(x) - 1}{[\ln(x)]^2}$.

The chain rule

If $y = f(g(x))$, let $u = g(x)$ so $y = f(u)$. Then

In words: differentiate the outside leaving the inside alone, then multiply by the derivative of the inside.

Example. Differentiate $y = \sin(3 x^2)$.

Inside: $u = 3 x^2$, $\frac{du}{dx} = 6 x$. Outside: $\frac{dy}{du} = \cos(u)$.

$\frac{dy}{dx} = \cos(3 x^2) \cdot 6 x = 6 x \cos(3 x^2)$.

Chain rule shortcuts for common composites

These are worth memorising as patterns:

• $\frac{d}{dx}(e^{f(x)}) = f'(x) e^{f(x)}$
• $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$ for $f(x) > 0$
• $\frac{d}{dx}(\sin(f(x))) = f'(x) \cos(f(x))$
• $\frac{d}{dx}(\cos(f(x))) = -f'(x) \sin(f(x))$
• $\frac{d}{dx}([f(x)]^n) = n [f(x)]^{n-1} f'(x)$

Combining rules

Many Paper 1 questions combine two or three rules.

Worked example. Differentiate $f(x) = x^2 e^{3 x}$.

Product rule with $u = x^2$, $v = e^{3 x}$. $u' = 2 x$, $v' = 3 e^{3 x}$ (chain rule on the inside).

$f'(x) = 2 x \cdot e^{3 x} + x^2 \cdot 3 e^{3 x} = e^{3 x} (2 x + 3 x^2) = x e^{3 x} (2 + 3 x)$.

Worked example. Differentiate $f(x) = \ln(\cos(x))$.

Chain rule with outside $\ln$ and inside $\cos(x)$.

$f'(x) = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\frac{\sin(x)}{\cos(x)} = -\tan(x)$.

Examples in context

Example 1. Rate of change of a damped oscillation. A spring's displacement is $s(t) = e^{-t}\sin(t)$. Using the product rule with $u = e^{-t}$ ($u' = -e^{-t}$) and $v = \sin t$ ($v' = \cos t$): $s'(t) = -e^{-t}\sin t + e^{-t}\cos t = e^{-t}(\cos t - \sin t)$. The velocity is zero when $\cos t = \sin t$, i.e. $\tan t = 1$, first at $t = \frac{\pi}{4}$.

Example 2. Marginal cost from a power model. A firm's cost is $C(x) = 50\sqrt{x} = 50x^{1/2}$ thousand dollars for $x$ hundred units. The marginal cost is $C'(x) = 50 \times \frac{1}{2}x^{-1/2} = \frac{25}{\sqrt{x}}$. At $x = 25$, $C'(25) = \frac{25}{5} = 5$ thousand dollars per extra hundred units, showing the rational power rule in use.

Try this

Q1. Differentiate $y = (2x + 1)^5$. [2 marks]

• Cue. Chain rule: $y' = 5(2x + 1)^4 \times 2 = 10(2x + 1)^4$.

Q2. Differentiate $f(x) = x^2 \sin x$. [3 marks]

• Cue. Product rule: $f'(x) = 2x\sin x + x^2\cos x$.

Q3. Differentiate $y = \dfrac{e^x}{x}$. [3 marks]

• Cue. Quotient rule: $y' = \dfrac{e^x \cdot x - e^x \cdot 1}{x^2} = \dfrac{e^x(x - 1)}{x^2}$.