Question

How do you differentiate `f(x)= x * (4-x^2)^(1/2) *ln x` using the product rule?

Answer 1

`f'(x)=-(2x^2-4)/(4-x^2)^(1/2)`

Explanation:

`"given "f(x)=g(x)h(x)" then "`

`f'(x)=g(x)h'(x)+h(x)g'(x)larr" product rule"`

`g(x)=xrArrg'(x)=1`

`h(x)=(4-x^2)^(1/2)larr" differentiate using "color(blue)"chain rule"`

`rArrh'(x)=1/2(4-x^2)^(-1/2)xxd/dx(4-x^2)`

`color(white)(rArrh'(x))=-x(4-x^2)^(-1/2)`

`rArrf'(x)=x(-x(4-x^2)^(-1/2))+(4-x^2)^(1/2)`

`color(white)(rArrf'(x))=(4-x^2)^(-1/2)(-x^2+4-x^2)`

`color(white)(rArrf'(x))=-(2x^2-4)/(4-x^2)^(1/2)`