# Derivative Question- Chain Rule

1. May 16, 2010

### char808

1. The problem statement, all variables and given/known data

The derivative of the function
h(x) = sin((x2 + 1)2)

2. Relevant equations

Chain Rule

3. The attempt at a solution

h(x) = sin((x2 + 1)2)

f(u) = sinu^2, f'(u)= 2ucosu^2
g(x) = x^2+1 g(x)= 2x

I get lost putting this back together but:

2(sinu^2)[cos(sinu^2)^2](2x) ?

2. May 16, 2010

### pbandjay

Are you trying to differentiate h(x) = sin [(x2 + 1)2] ? I'm currently having a hard time following your procedure (with the switching around of the h's and f's and u's and g's).

Last edited: May 16, 2010
3. May 16, 2010

### Staff: Mentor

This apparently is h(x) = sin((x2 + 1)2).
My guess is that your are looking at a formula for the chain rule as h(x) = f(g(x)).
In your problem, f(u) = sin(u), not sin2(u), so f'(u) = cos(u).

g(x) = ((x2 + 1)2), so g'(x) = ??

4. May 17, 2010

### char808

Sorry, I was a bit out of it last night.

h(x) = sin(x^2+1)^2

I have been taught to break this into f(u) and g(x) to apply chain rule

So:

f(u) = sin(u) f'(u)=cos(u)
g(x) = (x^2+1)^2 g'(x) = 4x(x^2+1)

h'(x) = cos((x^2+1)^2)4x(x^2+1)

5. May 17, 2010

### FieldDuck

Yep that seems right.

Just another note, I understand what you're doing, but I think if you end up confusing yourself by changing variables then it might not be worth it. I always thought of the chain rule as "Derivative of the outside function * derivative of inside function."

6. May 17, 2010

### char808

I conceptually could not grasp the chain rule last night. Today it "clicked" when I thought about it in those terms.

7. May 18, 2010

### Staff: Mentor

Part of the difficulty with this problem, from our side, was trying to understand what the function was.

In your first post, you had h(x) = sin((x2 + 1)2).

Later, you wrote this as h(x) = sin(x^2+1)^2. This is still somewhat ambiguous, as it is not clear exactly what is being squared. From you derivative, it seems to be this:
h(x) = sin((x^2+1)^2).

In the latter form it is clear that x --> x^2 + 1 --> (x^2 + 1)^2 --> sin((x^2 + 1)^2).

Part of being able to get help in this or other forums or other resources is being able to write your question clearly and unambiguously.