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

##### 1 Answer

Sep 26, 2017

#### Answer:

#### 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)#