Differentiating: Clarifying the Process

[DiamondV]

Homework Statement

Differentiate

Homework Equations

The Attempt at a Solution

du/dx = cos(x) dv/du=cos(u) dg/dv=cos(v)

dg/dx = dg/dv.dv/du.du/dx
= cosx.cos(sinx).cos(sin(sinx))

I know the answer is correct but my issue is in the understanding of the solution given. I understand it all except the u = sin(x). The approach I take is to work from the inside to the outside by finding u as the inner most and g(v) as the outermost. What I don't understand is why isn't u = x? isn't that the inner most?

Thanks.

 

[Mark44]

Homework Statement

Differentiate

Homework Equations

The Attempt at a Solution

du/dx = cos(x) dv/du=cos(u) dg/dv=cos(v)

dg/dx = dg/dv.dv/du.du/dx
= cosx.cos(sinx).cos(sin(sinx))

I know the answer is correct but my issue is in the understanding of the solution given. I understand it all except the u = sin(x). The approach I take is to work from the inside to the outside by finding u as the inner most and g(v) as the outermost. What I don't understand is why isn't u = x? isn't that the inner most?

You should never make the substitution u = x, because it's not useful. In that case, all you're doing is changing the name of the variable from x to u. The innermost function of x is sin(x).

 

[DiamondV]

You should never make the substitution u = x, because it's not useful. In that case, all you're doing is changing the name of the variable from x to u. The innermost function of x is sin(x).

Ah. So I should think of it as the innermost FUNCTION of x not just x. So its all about the functions? What if it was x^2 instead of the x? would the innermost function be x^2?

 

[Mark44]

Ah. So I should think of it as the innermost FUNCTION of x not just x. So its all about the functions? What if it was x^2 instead of the x? would the innermost function be x^2?

Yes, because this is a function of x, rather than just plain x.

 

[DiamondV]

Yes, because this is a function of x, rather than just plain x.

Perfect. Thanks so much!