Chain Rule for Nested and Multivariable Functions

All flashcards

Chain rule for a composition

For $h(x)=f(g(x))$, $h'(x)=f'(g(x))g'(x)$; evaluate the outer derivative at the inner output.

Differentiating multiple nested layers

Differentiate each successive link in the dependency chain, evaluate at its immediate input, and multiply the factors.

Multivariable chain rule for one input

If $y=f(u_1,\ldots,u_m)$ and each $u_\ell$ depends on $x$, then $\frac{dy}{dx}=\sum_{\ell=1}^{m}\frac{\partial y}{\partial u_\ell}\frac{du_\ell}{dx}$.

Second derivative of a composition

For $y=f(g(x))$, $y''=f''(g(x))(g'(x))^2+f'(g(x))g''(x)$.

Jacobian chain rule

For a composition $f\circ g$, $J_{f\circ g}(a)=J_f(g(a))J_g(a)$.

Quiz questions

1. If $h(x)=f(g(x))$, which expression gives $h'(x)$?Answer: $f'(g(x))g'(x)$
2. If $y=f(u_1,\ldots,u_m)$ and every $u_\ell$ depends on $x$, how is $dy/dx$ calculated?Answer: $\sum_{\ell=1}^{m}\frac{\partial y}{\partial u_\ell}\frac{du_\ell}{dx}$
3. For differentiable functions $f$ and $g$, with $g(x)\ne 0$, which formula gives the derivative of $f(x)/g(x)$?Answer: $\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$
4. If $f$ is the differentiable inverse of $g$, which formula gives $f'(y)$ when $g'(f(y))\ne 0$?Answer: $\frac{1}{g'(f(y))}$
5. If $f$ maps intermediate variables to outputs and $g$ maps inputs to intermediate variables, what is the Jacobian of $f\circ g$ at $a$?Answer: $J_f(g(a))J_g(a)$