Derivative Question- Chain Rule

[char808]

Homework Statement

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

Homework Equations

Chain Rule

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)= 2xI get lost putting this back together but:

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

 

[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).

 

[Mark44]

Homework Statement

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

This apparently is h(x) = sin((x² + 1)²).

Homework Equations

Chain Rule

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

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) = ??

I get lost putting this back together but:

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

 

[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)

 

[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."

 

[char808]

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

 

[Mark44]

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.