Differentiation of one function with respect to another function

Differentiation of one function with respect to another function :

If y = f (x) is differentiable, then the derivative of y with respect to x is

If f and g are differentiable functions of x and if

Example 10.28

Find the derivative of xx  with respect to x log x.

Solution

Example 10.29

Find the derivative of tan−1 (1+ x2 ) with respect to x2 + x +1.

Solution :

Let f (x) = tan−1 (1+ x2 )

g(x) =  x2 + x +1

Example 10.30

Differentiate sin( ax2 + bx + c) with respect to cos(lx 2 + mx + n)

Solution

Let u = sin( ax2 + bx + c)

v = cos(lx2 + mx + n)