Differentiate inverse trig functions, use implicit differentiation, and integrate rational functions, partial fractions and trig forms

What this dot point is asking

SCSA Unit 3 extends calculus in three directions: the derivatives of the inverse circular functions, implicit differentiation (including related rates), and a wider set of integration techniques. You must move between differentiation and integration fluently, since most integration here is "differentiation run backwards".

Derivatives of inverse trigonometric functions

For a scaled argument, combine with the chain rule. For example $\dfrac{d}{dx}\arcsin\!\big(\tfrac{x}{a}\big) = \dfrac{1}{\sqrt{a^2 - x^2}}$ and $\dfrac{d}{dx}\arctan\!\big(\tfrac{x}{a}\big) = \dfrac{a}{a^2 + x^2}$.

Implicit differentiation

When a relation links $x$ and $y$ without $y$ being isolated (for example $x^2 + y^2 = 25$), differentiate both sides with respect to $x$, treating $y$ as a function of $x$. Every $y$ term gains a factor $\dfrac{dy}{dx}$ from the chain rule, then you solve algebraically for $\dfrac{dy}{dx}$.

Related rates

Related-rates problems link two changing quantities through a known equation and use the chain rule $\dfrac{dA}{dt} = \dfrac{dA}{dx}\cdot\dfrac{dx}{dt}$. Identify the variables, write the relation, differentiate with respect to time, then substitute the instantaneous values last.

Integration techniques

Integration in Unit 3 is built on recognition and substitution. Key standard forms (with $a > 0$ and an arbitrary constant $C$):

The form $\dfrac{f'(x)}{f(x)}$ is the workhorse for rational integrands: if the numerator is (a multiple of) the derivative of the denominator, the integral is a logarithm. Otherwise, use a $u$-substitution or rearrange the numerator to manufacture $f'(x)$.

When the integrand splits into separate fractions (for instance a numerator $x + 1$ over $x^2 + 4$), break it into a logarithmic piece $\dfrac{x}{x^2+4}$ and an inverse-tangent piece $\dfrac{1}{x^2+4}$ and integrate each separately.