The Chain Rule and Implicit Differentiation

Many functions met in practice are composite functions — one function applied to the output of another, such as $y=(3x^2+5)^4$, which is 'raise-to-the-4th' applied to '$3x^2+5$'. Differentiating such a function directly from first principles, or by expanding it out, is usually impractical. The chain rule handles composites directly.

If $y=f(u)$ and $u=g(x)$, so that $y=f(g(x))$ is a function of $x$ via the intermediate variable $u$, then

In words: differentiate the 'outer' function with respect to its own variable $u$, differentiate the 'inner' function with respect to $x$, and multiply the two results together.

For $y=(3x^2+5)^4$: let $u=3x^2+5$, so $y=u^4$. Then $\dfrac{dy}{du}=4u^3$ and $\dfrac{du}{dx}=6x$, giving $\dfrac{dy}{dx} = 4u^3 \cdot 6x = 24x(3x^2+5)^3$.

Implicit differentiation. Sometimes $y$ and $x$ are tied together by an equation that is not (and cannot easily be) solved to give $y$ explicitly as a function of $x$ — e.g. $x^2+y^2=25$. We can still find $\dfrac{dy}{dx}$ by differentiating both sides of the equation with respect to $x$, treating $y$ throughout as an unknown function of $x$ and applying the chain rule whenever a term involves $y$ (so that $\dfrac{d}{dx}(y^n) = n y^{n-1}\dfrac{dy}{dx}$, not simply $ny^{n-1}$). After differentiating, we collect every term containing $\dfrac{dy}{dx}$ on one side and solve for it algebraically. …