How Do You Apply the Chain Rule to Differentiate f(x) = x^5(4^(x^2))?

[ninanana]

I'm so confused. I have to find the derivative of f(x) = x^5(4^(x^2)). All of the powers are messing me up. Any help would be much appreciated. Thanks!

 

[hypermorphism]

Is your function $$f(x) = x^5(4^{x^2})$$ or $$f(x) = x^{5(4^{x^2})}$$ ?

 

[ninanana]

The first one, sorry.

 

[Karlsen]

Use product rule first, then you end up differentiating 4^(x^2).
A nice formula to know is d/dx ( a^(f(x)) ) = a^f(x) * ln(a) * f'(x), which comes from the chain rule.

 

[ninanana]

I know I'm having some sort of stupid lapse right now, but the part I can't figure out is the 4^(x^2).

 

[HallsofIvy]

To differentiate $$y= 4^{x^2}$$, take the logarithm of both sides:
$$ln y= x^2 ln 4$$
Now differentiate that, with respect to x.
$$\frac{1}{y}y'= 2x ln 4$$
so
$$y'= 2x (ln 4)y= 2x 4^{x^2} ln 4$$

Karlsen used the fact that the derivative of a^x is a^x ln a, but not everyone knows that!