Question

Can someone please help me find the derivative of the following function?

Answer 1

\[ y=\cos^7(\sin10x)\]

Answer 2

How do I go about finding the derivative?

Answer 3

\[ y'=\cos^7(\frac{ d}{ dx }\sin[10x])+(\sin[10x])(\frac{ d }{ dx }\cos^7)\]

Answer 4

Am I on the right track?

Answer 5

the 7 is an exponent right?

Answer 6

if so you use the chain rule

Answer 7

avoid the product rule altogether?

Answer 8

i think so

Answer 9

i'm not thinking straight today - i'll have another look at this

Answer 10

no worries, thanks

Answer 11

ok
let u = sin 10x
du/dx = 10 cos10x
y = cos^7 u
dy/du = -7 sin^6 u = -7 (sin sin10x)^6

dy/dx = -70 cos 10x (sin sin 10x)^6

Answer 12

I would first write it out like (cos(sin10x))^7 to clearly see the 'outside' and 'inside' functions for chain rule application. So, stepwise:
d/dx of (cos(sin 10x))^7
=7(cos(sin 10x))^6 * d/dx of cos(sin 10x)
= .......................... * -sin(sin 10x) * d/dx of sin (10x)
= .......................... * .................... * cos(10x) * d/dx of (10x)
= .......................... * .................... * ............. * 10
Then putting that all together gives
7(cos(sin 10x))^6 * -sin(sin 10x) * cos(10x) * 10
You may do some simplification if you want to. I hope that helps.