Finding the Derivative of a Complex Exponential Function

[Seb97]

Homework Statement

Find f', where f(x) = 1+x^2)^(x^2)

Homework Equations

The Attempt at a Solution

I attempted the question using the chain rule but I was told that you do not use the. That your meant to use logs. But I have no idea where to begin. Any help or tips would be much appreciated.

 

[mg0stisha]

Have you learned logarithmic differentiation yet?

 

[Seb97]

Hi mg0stisha

Ya I am pretty sure we touched on it but I was convinced I had to use the chain rule to solve this.

 

[nobahar]

Hello!

There may be an easier way, but take the ln of both sides. This way, you will need to be able to do a little manipulation of logs, to know the product rule of differentiation and to know the chain rule. Remember, if you take the lns of both sides, you will need the chain rule, because you will need to find the derivative of the natural log of the function with respect to the function, and then the derivative of the function with respect to x, which is what you are looking for:

So, if:

$$(1+x^2)^{x^2} = y$$

then when you take the natural log of both sides, you end up with ln y. So the derivative with respect to x (the right hand side of the equation only) is:

$$\frac{d}{dy} \left \left ln{y} \left \left * \left \left \frac{dy}{dx}$$

I didn't include the left hand side, as I figured you might want to have a go yourself! Someone may be able to offer better advice.

I hope that helps!

 

[Seb97]

Hey mg0stisha

Thank you for the reply. It was quite helpful and thank you for not providing me with the full solution. I think I got cheers