Inverse of a derivative: Let f(x) = 10^{pi} + int_x^{-x^3} sqrt{1 + t^8} dt...

[Awesomeiam]

I can not figure out how to do part c. I’ve tried to take the derivative of both sides but it doesn’t work out, and I can’t solve the integral either. Could I get some help.

 

[khansaheb]

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I can not figure out how to do part c. I’ve tried to take the derivative of both sides but it doesn’t work out, and I can’t solve the integral either. Could I get some help.

You have to use Leibniz's theorem - have you studied the theorem?

 

[Dr.Peterson]

I can not figure out how to do part c. I’ve tried to take the derivative of both sides but it doesn’t work out, and I can’t solve the integral either. Could I get some help.

I'd use the inverse function theorem as taught here, for instance:

3.7: Derivatives of Inverse Functions

The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. We can use the inverse function theorem to develop …

math.libretexts.org

For a start, you need to find [imath]f^{-1}(10^\pi)[/imath]. Do you see how to find it? For what x is [imath]f(x)=10^\pi[/imath]?