Differentiation: Composite, Implicit, and Inverse Functions
What you must understand
The chain rule differentiates compositions: $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$. Layer derivatives from outside in, and expect nested chains. Combine with product and quotient rules for mixed expressions.
Implicit differentiation handles relations not solved for $y$: differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (so $\frac{d}{dx}y^2=2y\frac{dy}{dx}$), then solve for $\frac{dy}{dx}$. This gives tangent slopes on curves like circles and ellipses.
For inverses, $\frac{d}{dx}f^{-1}(x)=\frac{1}{f'(f^{-1}(x))}$, and at a point, slopes of inverse functions are reciprocals. Know derivatives of inverse trig functions such as $\arcsin$, $\arctan$. Logarithmic differentiation helps with variable exponents like $x^x$.
Traps
- Omitting the inner derivative in the chain rule.
- Forgetting the $\frac{dy}{dx}$ factor when differentiating $y$ terms implicitly.
- Plugging the wrong value into the inverse derivative formula (evaluate $f'$ at the inverse's output).
- Treating $\frac{d}{dx}(y)$ as zero when $y$ depends on $x$.
Worked example
Differentiate $y=\sin(x^2+1)$. Outer derivative is $\cos(x^2+1)$; inner is $2x$. Thus $\frac{dy}{dx}=2x\cos(x^2+1)$.
Rule of thumb: for inverses, find the input to $f'$ by solving $f(a)=$ the given output, then take the reciprocal slope.