Related Rates and Implicit Differentiation

Implicit Differentiation

Not every curve is written as $y = f(x)$. An implicit relation like $x^2 + y^2 = 25$ defines $y$ through $x$ without solving for it explicitly.

Implicit differentiation finds $\frac{dy}{dx}$ by differentiating both sides with respect to $x$. Treat $y$ as a function of $x$ and apply the chain rule to every $y$-term.

Solving yields $\frac{dy}{dx} = -\frac{x}{y}$. This handles tangled curves that resist explicit isolation.

Related Rates

When quantities link up and change over time $t$, their rates link up too. Differentiate with respect to $t$, bringing in a $\frac{d}{dt}$ factor for every changing variable.

Method: Write equation $\rightarrow$ Differentiate wrt $t$ $\rightarrow$ Substitute and solve.

For a sphere ($V = \frac{4}{3}\pi r^3$), differentiating gives $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

Common pitfall: Forgetting the chain rule factor (like $\frac{dy}{dx}$ or $\frac{dr}{dt}$) on dependent variables. Treating $y^2$ as just $2y$ is the classic fatal error.