change_of_variables_diff.md

tags
math

Change of variables in differentiation

When differentiating we can use the chain rule to find the correct substitution. Consider $f^\prime(g(x))$ and let's say we substitute $y := g(x)$, then:

$$ \begin{aligned} \frac{\partial f(g(x))}{\partial x} &= \frac{\partial f(g(x))}{\partial g(x)} \frac{\partial g(x)}{\partial x} \\ &= \frac{\partial f(y)}{\partial y} \frac{\partial y}{\partial x} \end{aligned} $$

The $\partial y/\partial x$ is the 'extra' part we have to add for the substitution to be valid.

Example

For example, for $y:= x^2$:

$$ \frac{\partial \sin x^2}{\partial x} = \frac{\partial \sin y}{\partial y} 2x = (\cos y) 2x = 2x \cos x^2 $$