Hello. I am seeking help solving the problem below. I think I know how to solve it, but I'm hoping I can get a second set of eyes. Thanks.

Question

Given f(x) = ( sin (5x) ) ^x , find f ' (x) .

I believe I would use this rule: e ^ ( ln(x) ) = x . Is this correct?

If so, would I then do this: e ^ ( ln(sin (5x) ) ) ? What would I do afterwards?

I may have began this problem the wrong way, so if you wouldn't mind giving me the steps I would appreciate that immensely.

Thank you, and enjoy the rest of your day.

Jackson

Answer 1

Here is a video explanation. Please let me know if you have any questions or need clarification.

Answer 2

Here are some hints to get you started. Let me know if you need additional clarification.

Let y = (sin(5x)^x

Take the natural log of both sides: ln y = ln (sin(5x))^x

Apply properties of logarithms to get ln y = x ln (sin(5x))

Now take the derivative of both sides with respect to x. The left side will become y'/y. on the right side you will have to use the product rule.

In the end you will have to multiply both sides by y to isolate y'

Answer 3

y = (sin 5x)^x

ln y = x ln (sin 5x)

Differentiate implicitly

1/y dy/dx = ln (sin 5x) + x. (1/sin 5x) (5 cos 5x)

dy/dx = y [ ... ] = (sin 5x) ^x [ .... ]