Let `f(x)= (8x^5)(sqrt(x))+((4)/((x^2)(sqrt(x))))` Find `f '(x)` .  

Expert Answers
lemjay eNotes educator| Certified Educator

`f(x) = 8x^5sqrtx+ 4/(x^2sqrtx)`

To simplify f(x), express the radicals as exponents.  Note that `sqrtx=x^(1/2)`.

`f(x) = 8x^5*x^(1/2) + 4/(x^2*x^(1/2))`

Then, use the rule of exponents for multiplying same base which is `x^m*x^n=x^(m+n)` .

`f(x) = 8x^(11/2) + 4/x^(5/2)`

`f(x)=8x^(11/2) + 4x^(-5/2)`

Then, let's take the derivative of f(x).

`f'(x) = d/(dx) 8x^(11/2) + d/(dx) 4x^(-5/2)`

To determine d/dx of each term, use the power rule formula of derivatives which is `d/(du)cu^n = c*n u^(n-1)` .

>>  `d/(dx) 8x^(11/2) = 8*11/2x^(9/2)=44x^(9/2)= 44x^4x^(1/2)= 44x^4sqrtx`

>> `d/(dx)4x^(-5/2)= 4*(-5/2)x^(-7/2)=-10/x^(7/2)=-10/(x^3sqrtx)`

So we have,

`f'(x) = 44x^4sqrtx-10/(x^3sqrtx)`


Hence, the derivative of `f(x)=8x^5sqrtx + 4/(x^2*sqrtx)` is 

`f'(x) = 44x^4sqrtx - 10/(x^3sqrtx)` .