find f'(x) of f(x)=sin(2x)(sqrt(1+cos(5x)))

Question

misty1212 · Answer

HI!!

misty1212 · Answer

product plus chain rule 
$$(fg)'=f'g+g'f$$ with $$f(x)=\sin(x), f'(x)=\cos(x), g(x)=\sqrt{1+\cos(5x)}$$
$$g'(x)=\frac{-5\sin(5x)}{2\sqrt{1+\cos(5x)}}$$

clara1223 · Answer

what happened to the 2x inside sin(x)?

misty1212 · Answer

oops $$f(x)=\sin(2x), f'(x)=2\cos(2x)$$ my bad

clara1223 · Answer

How would I apply the chain rule to that? The sqrt makes it pretty messy.

clara1223 · Answer

@misty1212 ?

misty1212 · Answer

$$\left(\sqrt{f}\right)'=\frac{f'}{2\sqrt{f}}$$

misty1212 · Answer

which should explain $$g'(x)=\frac{-5\sin(5x)}{2\sqrt{1+\cos(5x)}}$$

clara1223 · Answer

That makes sense. I'm just not sure what to do from there. From the product rule I got 2cos(2x)(sqrt(1+cos(5x)))+sin(2x)((-5sin(5x))/2(sqrt(1+cos(5x))))

misty1212 · Answer

leave it

clara1223 · Answer

that's my final answer?

misty1212 · Answer

it is a silly made up question anyway
what else can you do?

misty1212 · Answer

you sure as hell don't want to actually add them although you could if you had like an extra half hour to waste

clara1223 · Answer

Haha definitely not. Still got 6 more questions on this homework sheet due tomorrow.

misty1212 · Answer

best get busy

misty1212 · Answer

lol not that "get busy"