Question

How do I solve f'(x) for 3sqrt(t)+(11/sqrt(t))?

I started trying to use the power rules... 3t^(1/2)+(11)t^(-1/2) --> 1.5t(-1/2)+(11^(-1/2))(t^(3/2))... but I get stuck after that last part (and I'm pretty sure I'm not doing this right too. Can someone help me figure out how to find f'(x) of this? Please provide lots of work so I can follow along and do similar problems myself. Thanks so much!!

Answer 1

`f'(t)=3/2t^(-1/2)-11/2t^(-3/2)`

Explanation:

We want to find the derivative of

To use the power rule, we want the form `f(x)=color(red)(a)x^(color(blue)(n))`

Rewrite using the exponential rules

`f(t)=3sqrt(t)+11/sqrt(t)`

`f(t)=color(red)(3)t^(color(blue)(1/2))+color(red)(11)t^(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'(t)=color(blue)(1/2)color(red)(3)t^(color(blue)(1/2)-1)+(color(blue)(-1/2))color(red)(11)t^(color(blue)(-1/2)-1)`

`f'(t)=3/2t^(-1/2)-11/2t^(-3/2)`