Derivative of a composite function?

[potmobius]

Homework Statement

Find the derivative of the function: cos(x)^(cos(cos(x)))

Homework Equations

The chain rule

The Attempt at a Solution

I know how the chain rule works, and I've done many problems with composite functions. However, I just don't know where to start with this one. I'm lost and confused :(

 

[Elucidus]

Correct me if I am wrong. It seems that you are trying to find

$$\frac{d}{dx}\;(\cos x)^{\cos(\cos x)}$$

Any time you need to differentiate an expression that involves a variable base and exponent (as we have here) you need to use logarithms and implicit differentiation.

i.e.

$$y=f(x)^{g(x)}$$

$$\ln y = \ln \left (f(x)^{g(x)} \right)$$

$$\ln y = g(x) \cdot \ln (f(x))$$

$$\frac{d}{dx} \ln y = \frac{d}{dx} \left[ g(x) \cdot \ln (f(x)) \right]$$

$$\frac{y'}{y} = \frac{g(x) \cdot f'(x)}{f(x)} + g'(x) \cdot \ln (f(x))$$

$$y' = y \cdot \left[ \frac{g(x) \cdot f'(x)}{f(x)} + g'(x) \cdot \ln (f(x)) \right]$$

$$y' = \left( f(x)^{g(x)} \right) \cdot \left[ \frac{g(x) \cdot f'(x)}{f(x)} + g'(x) \cdot \ln (f(x)) \right]$$

Hopefully this will get you going in the right direction.

--Elucidus

 

[Dick]

I'm glad to hear you know how the chain rule works. Now prove it. cos(x) is exp(log(cos(x)). Does that help? Now use properties of exponents and the chain rule. You'll need some product rule as well.

 

[potmobius]

Yes, that's exactly what I meant, and this clears it up! Thanks :)