Differentiating trig, exponential and log functions

The standard derivatives

Level 3 adds derivatives beyond polynomials. The ones you must know:

$\frac{d}{dx}(\sin x) = \cos x \qquad \frac{d}{dx}(\cos x) = -\sin x \qquad \frac{d}{dx}(\tan x) = \sec^2 x$

$\frac{d}{dx}(e^x) = e^x \qquad \frac{d}{dx}(\ln x) = \frac{1}{x}$

Note $e^x$ is the only function that is its own derivative.
For trig functions, $x$ must be in radians.

Combining with the chain rule

When the inside is more than just $x$ , use the chain rule — differentiate the standard function, then multiply by the derivative of the inside:

$\frac{d}{dx}\big(\sin(kx)\big) = k\cos(kx) \qquad \frac{d}{dx}\big(e^{kx}\big) = k\,e^{kx} \qquad \frac{d}{dx}\big(\ln(kx)\big) = \frac{1}{x}$

Worked ExampleChain rule with e and trig

Differentiate (a) $y = e^{3x}$ and (b) $y = \sin(2x)$ .

Part (a) — exponential with the chain rule

The inside is $3x$ , whose derivative is $3$ :

$\frac{dy}{dx} = e^{3x} \times 3 = 3e^{3x}$

Part (b) — trig with the chain rule

The inside is $2x$ , whose derivative is $2$ :

$\frac{dy}{dx} = \cos(2x) \times 2 = 2\cos(2x)$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Differentiate $y = e^{5x}$ .

Merit

Differentiate $y = 3\cos(4x) + \ln x$ .

Excellence

Differentiate $y = e^{2x}\sin x$ .