Differentiation of a function w.r.t another function

So far we have discussed derivatives of one variable, say y with respect to other variable, say , x . in this section we will discuss derivative of a function with respect to another function.Let $\displaystyle u=f\left( x \right)and\quad v=g\left( x \right)$ be two functions of x. Then to find the derivative of f(x) w.r.t. g(x) i.e., to find $\displaystyle \frac { du }{ dv }$ we use the following formula

$\displaystyle \frac { du }{ dv } =\frac { \frac { du }{ dx } }{ \frac { dv }{ dx } }$

Thus, to find the derivative of f(x) w.r.t. g(x) we differentiate both w.r.t. x and then divide the derivative of f(x) w.r.t. x by the derivative of g(x) w.r.t. x.

Example

Differentiate $\displaystyle \log { \sin { x } w.r.t.\sqrt { \cos { x } } }$

Solution:

$\displaystyle u=\log { \sin { x } and\quad v=\sqrt { \cos { x } } } then\\ \frac { du }{ dx } =\cot { x } \quad and\quad \frac { dv }{ dx } =-\frac { \sin { x } }{ 2\sqrt { \cos { x } } } \\ \therefore \frac { du }{ dv } =\frac { \frac { du }{ dx } }{ \frac { dv }{ dx } } \\ =\frac { \cot { x } }{ -\frac { \sin { x } }{ 2\sqrt { \cos { x } } } } \\ =-2\sqrt { \cos { x } } .\cot { x } .cosec\quad x$