Year 12 Calculus: Logarithmic and trigonometric differentiation

Logarithmic and trigonometric functions introduce new derivative rules, but the process is still the same: identify the function type first, then decide whether a rule such as the chain rule is also needed.

Know the base rules first:

d (\ln x) / d x = 1 / x

d (\sin x) / d x = \cos x

and

d (\cos x) / d x = - \sin x

Worked example

Problem: Differentiate

y = \ln (x^{2} + 1) + 3 \sin x

Differentiate $\ln (x^{2} + 1)$ using the chain rule: $d / d x = (1 / (x^{2} + 1)) * 2 x = 2 x / (x^{2} + 1)$ .
Differentiate $3 \sin x$ : $d / d x = 3 \cos x$ .
Add the derivatives: $d y / d x = 2 x / (x^{2} + 1) + 3 \cos x$ .

Answer:

d y / d x = 2 x / (x^{2} + 1) + 3 \cos x

Rule selection matters

These functions often combine several derivative ideas at once. A logarithm may also need the chain rule. A trigonometric term may be part of a product or quotient. The best habit is to identify the outermost structure first and then decide whether any inner structure needs another rule.

Worked example 2

Problem: Differentiate

y = \cos (3 x) + \ln (2 x - 5)

Differentiate $\cos (3 x)$ using the chain rule: $d / d x = - \sin (3 x) x 3 = - 3 \sin (3 x)$ .
Differentiate $\ln (2 x - 5)$ using the chain rule: $d / d x = 1 / (2 x - 5) x 2 = 2 / (2 x - 5)$ .
Add the results: $d y / d x = - 3 \sin (3 x) + 2 / (2 x - 5)$ .

Answer:

d y / d x = - 3 \sin (3 x) + 2 / (2 x - 5)

Common mistakes

Writing the derivative of $\cos x$ as $\sin x$ instead of $- \sin x$ .
Forgetting the inner derivative in $\ln (2 x - 5)$ or $\sin (3 x)$ .
Dropping brackets in fraction derivatives.

This part of calculus feels difficult mainly because several rule families now overlap. A mixed function may require you to recognise the basic derivative rule, the chain rule, and the surrounding algebraic structure all at once. That is normal. The way to make it reliable is to slow down long enough to identify the function type before any differentiation begins.

When revising, say the derivative rule aloud before using it. For example, "the derivative of $\ln u$ is $u' / u$ " or "the derivative of $\cos u$ is $- \sin u$ times $u'$ ". Speaking the rule in words often makes sign errors and missing chain-rule factors much easier to catch.

Revision focus

In calculus, one of the best habits is to decide what role the expression is playing before you touch a rule. Is it a product, a composite function, an implicitly defined curve, or an antiderivative problem? That classification step is not wasted time. It is the step that usually determines whether the next line of working will be short and accurate or long and unstable. Strong calculus work looks calm because the structure is identified early, not because the algebra is magically easier.