differentiate

Question

the function              y=a^f(x)

Answer 1

Assuming that "a" is a constant, to differentiate y = a^f(x) we use logarithmic differentiation.

 

lny = ln[a^f(x)]

 

      = f(x)lna

 

So, (1/y)(dy/dx) = f'(x)lna + f(x)(0)

 

                  dy/dx = yf'(x)lna

 

                              = a^f(x) f'(x) lna

 

Example:  y = 2^sinx    Then, dy/dx = 2^sinx cosx ln2

Answer 2

y = a^f(x)

 

 

We apply logarithmic differentiation. Log both sides of equation.

 

lny = f(x)lna

 

 

Differentiate both sides of equation using implicit differentiation.

 

1 / y = d/dx[f(x)]lna + f(x)(1/a)

 

 

Next, solve for d/dx[f(x)] from this equation.