Differentiate sine, cosine and tangent functions and their compositions via the chain rule

What this dot point is asking

VCAA wants you to differentiate sine, cosine and tangent functions and their composites using the chain rule.

Standard derivatives

These derivations use the small-angle limit $\lim_{\theta \to 0} \sin\theta/\theta = 1$ together with the addition formulas. The cosine result picks up the minus sign because $\cos\theta$ decreases as $\theta$ moves from zero.

Chain rule extensions

Standard combinations

Trig times polynomial

Product rule. $\frac{d}{dx} (x \sin x) = \sin x + x \cos x$.

Squared trig

Chain rule with $u^2$. $\frac{d}{dx} \sin^2 x = 2 \sin x \cos x = \sin 2x$.

Trig over polynomial

Quotient rule.

Connection to oscillation

If $x(t) = A \sin(\omega t)$ is the displacement of a simple oscillator, then:

$v(t) = dx/dt = A \omega \cos(\omega t)$

$a(t) = dv/dt = -A \omega^2 \sin(\omega t) = -\omega^2 x(t)$

This is the differential equation of simple harmonic motion. The derivative pattern explains why oscillators have constant period independent of amplitude.

Choosing the right rule

Trig derivatives appear inside products, quotients and composites. A product such as $x\sin x$ uses the product rule; a power such as $\sin^2 x = (\sin x)^2$ uses the chain rule with outer power $u^2$; a quotient such as $\dfrac{\sin x}{x}$ uses the quotient rule; and a nested argument such as $\sin(x^2 + 1)$ uses the chain rule on the argument. When more than one rule applies, work from the outside in: identify the outermost operation, apply its rule, and differentiate the inner part as a sub-problem. Many VCAA items then ask you to set the derivative to zero, so simplifying with double-angle identities (for example $2\sin x\cos x = \sin 2x$) often makes the equation solvable by hand.

Finding tangents and stationary points

The derivative gives the gradient of the tangent at any point, so for $y = f(x)$ the tangent at $x = a$ has gradient $f'(a)$ and equation $y - f(a) = f'(a)(x - a)$. Stationary points occur where $f'(x) = 0$; for trig functions these recur periodically, so always restrict to the stated interval and report every solution in it. Because $\sin$ and $\cos$ oscillate, a trig function typically has alternating maxima and minima, which a sign test on $f'$ confirms.

In one sentence

The standard derivatives are $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$, $\frac{d}{dx} \tan x = \sec^2 x$, with chain-rule extensions $\frac{d}{dx} \sin u = u' \cos u$ and equivalents, assuming $x$ is in radians throughout.

Examples in context

Example 1. Velocity of an oscillating piston. A piston's displacement is $x(t) = 4\sin(2t)$ cm, with $t$ in seconds. Its velocity is $x'(t) = 4 \times 2\cos(2t) = 8\cos(2t)$ cm/s. The maximum speed is $8$ cm/s, occurring whenever $\cos(2t) = \pm 1$, that is at $t = 0, \frac{\pi}{2}, \pi, \dots$ (the centre of each swing).

Example 2. Slope of a tidal curve. A tide height is $h(t) = 3\cos\!\left(\frac{\pi}{6}t\right)$ m, $t$ in hours. The rate of change is $h'(t) = -3 \times \frac{\pi}{6}\sin\!\left(\frac{\pi}{6}t\right) = -\frac{\pi}{2}\sin\!\left(\frac{\pi}{6}t\right)$ m/h. At $t = 3$ (a quarter period), $h'(3) = -\frac{\pi}{2}\sin\!\left(\frac{\pi}{2}\right) = -\frac{\pi}{2} \approx -1.57$ m/h, the fastest fall.

Try this

Q1. Differentiate $f(x) = \cos(5x)$ and $g(x) = \tan(3x)$. [2+2 marks]

• Cue. $f'(x) = -5\sin(5x)$; $g'(x) = 3\sec^2(3x)$.

Q2. Find the derivative of $y = x\cos x$. [3 marks]

• Cue. Product rule: $y' = \cos x - x\sin x$.

Q3. A particle's displacement is $s(t) = 2\sin(3t)$ m. (a) Find the velocity $v(t)$. (b) Find the velocity at $t = 0$. [2+2 marks]

• Cue. (a) $v(t) = 6\cos(3t)$. (b) $v(0) = 6$ m/s.