Question

How do I solve f(x)=((5x^2)+(2x)+2)/(sqrt(x)) for f'(x)?

I'm still not understanding these power rules, and could definitely use some help. Thank you!

Answer 1

`f'(x)=(15x^(2)+2x-2)/(2x^(3/2))`

Explanation:

We want find the derivative of

`f(x)=(5x^2+2x+2)/sqrt(x)`

First rewrite so we easier can apply the power rule

`f(x)=(5x^2)/sqrt(x)+(2x)/sqrt(x)+2/sqrt(x)`

`f(x)=5x^(2-1/2)+2x^(1-1/2)+2x^(0-1/2)`

`f(x)=color(red)(5)x^(color(blue)(3/2))+color(red)(2)x^(color(blue)(1/2))+color(red)(2)x^(color(blue)(-1/2))`

Use the power rule if `f(x)=color(red)(a)x^(color(blue)(n))`

then `f'(x)=color(blue)ncolor(red)ax^(color(blue)(n)-1)`

Thus

`f'(x)=color(blue)(3/2)color(red)(5)x^(color(blue)(3/2)-1)+color(blue)(1/2)color(red)(2)x^(color(blue)(1/2)-1)+color(blue)((-1/2))color(red)(2)x^(color(blue)(-1/2-1))`

`color(green)(f'(x)=15/2x^(1/2)+x^(-1/2)-x^(-3/2))`

`f'(x)=(15x^(1/2))/2+1/(x^(1/2))-1/x^(3/2)`

`f'(x)=(15x^2)/(2x^(3/2))+x/(x^(3/2))-1/x^(3/2)`

`f'(x)=(15x^2)/(2x^(3/2))+(2x)/(2x^(3/2))-2/(2x^(3/2))`

`f'(x)=(15x^(2)+2x-2)/(2x^(3/2))`