Differentiation of One Function with Respect to Another Function

If $u=f(x)$ and $v=g(x)$ are both differentiable functions of the same variable $x$, the derivative of $u$ with respect to $v$ — meaning, treat $v$ as if it were the independent variable instead of $x$ — is defined as $\dfrac{du}{dv}=\dfrac{du/dx}{dv/dx}.$ This is exactly the parametric-derivative idea from section 1.4.2, but with $x$ itself playing the role of the parameter $t$: both $u$ and $v$ are thought of as riding along together as $x$ varies. To compute it: differentiate $u$ and $v$ separately with respect to $x$ in the ordinary way, then divide the two results.

Worked Example 1. Find the derivative of $7^x$ with respect to $x^7$. Let $u=7^x,\ v=x^7$; then $\dfrac{du}{dx}=7^x\log7$ and $\dfrac{dv}{dx}=7x^6$, so $\dfrac{du}{dv}=\dfrac{7^x\log7}{7x^6}$.

Worked Example 2. Find the derivative of $\cos^{-1}x$ with respect to $\sqrt{1-x^2}$. Let $u=\cos^{-1}x,\ v=\sqrt{1-x^2}$; then $\dfrac{du}{dx}=-\dfrac1{\sqrt{1-x^2}}$ and $\dfrac{dv}{dx}=-\dfrac{x}{\sqrt{1-x^2}}$, so $\dfrac{du}{dv}=\dfrac{-1/\sqrt{1-x^2}}{-x/\sqrt{1-x^2}}=\dfrac1x$. …