# Derivatives using chain rule

## Main Question or Discussion Point

id like some help deriving certain functions using the chain rule
the way our teacher does it is different from what the textbook says
he derives the outermost functions before getting to the innermost functions, this is where
i get confused =(

for example

f(x) = sincos(5x)
i get
f'(x) = (coscos(5x)) (-sin(5x)) (5)
= -5coscos(5x)sin(5x)

im not sure if this is right

another question:

f(x) = sin^4(2x) + cos^4(2x)

## Answers and Replies

nicksauce
Science Advisor
Homework Helper
The first looks correct... why not have a stab at the second?

here is what i get

f(x) = sin^4(2x) + cos^4(2x)

f'(x) = [ (4sin^3(2x) (cos^4(2x)) (2) ] + [ (4cos^3(2x) (-sin(2x)) (2) ]

= 8sin^3(2x)cos^4(2x) + -8cos^3(2x)sin(2x)

another question - this one i get a different answer than my friend

f(x) = [ ((3x^2 -2)^5) - 1 ] ^5

i get

f(x) = (3x^10 - 2^5 - 1)^5
= 3x^50 - 2^25 - 1^5

f'(x) = 150x^49

here is what im working on right now - ill post what i get in a few mins

f(x) = |x^1/3|
f(x) = x^2 cos^2(x^2)
f(x) = [ (2x+5)/(7x-9) ]^2/3

nicksauce
Science Advisor
Homework Helper
For the first one in your post you're not applying the chain rule correctly.

d/dx sin(2x)^4 = 4 * sin(2x) ^3 * d/dx (sin2x)

For your second one

(3x^10 - 2^5 - 1)^5 is certainly NOT the same as 3x^50 - 2^25 - 1^5

for f(x) = sin^4(2x) + cos^4(2x), i am doing it right im just deriving it wrong? i should derive each part then add together..if so would the answer be

8sin^3(2x)cos(2x) - 8cos^3(2x)sin(2x)

for f(x) = [ ((3x^2 -2)^5) - 1 ] ^5
i get

f'(x) = (150x)(3x^2 - 2)^24

another one im sketched on
f(x) = x^2 cos^2(x^2)

using product rule i get

f'(x) = (2x) (cos^2(2x)) + (x^2) (2cos(x^2)(sin(x^2))(2x)
=2xcos^x(x^2) + 4x^3cos(x^2)sin(x^2)