Find the derivative of the inverse of the function

f(x)=6x+(9x^21) when x=-15

I think you might have a typo

the power x^21 looks totally "out of character".

Anyway, I would write the function as
y = 6x + 9x^21

then its inverse is
x = 6y + 9y^21
differentiate implicitly with respect to x
1 = 6dy/dx + (189y^20)dy/dx
factoring and simplifying we get
dy/dx = 1/(6 + 189y^20)

so when x=-15 , in the inverse we get
-15 = 9y^21 + 6y
9y^21 + 6y + 15 = 0
I just noticed that y = -1 is a solution

so dy/dx = 1/(6 + (-1)^20)
= 1/(6+1) = 1/7

thnx, that helped alot!

To find the derivative of the inverse of a function, we can use the inverse function theorem. According to this theorem, if a function f is invertible and has a well-defined derivative at a point x, then the inverse function f^(-1) also has a well-defined derivative at the corresponding y = f(x).

In this case, we want to find the derivative of the inverse function of f(x) = 6x + 9x^21 with respect to y when y = f(x) = -15.

Step 1: Find the derivative of f(x) with respect to x.
Let's differentiate f(x) = 6x + 9x^21 to find its derivative:
f'(x) = 6 + 189x^20

Step 2: Find the value of x.
We know that y = f(x) = -15, so we need to find the corresponding value of x.
-15 = 6x + 9x^21

This equation is a polynomial equation, which can be difficult to solve algebraically. We can use numerical methods or approximation techniques to estimate the value of x.

Step 3: Compute the derivative of the inverse function.
Once we have found the value of x, we can calculate the derivative of the inverse function at that point.

If we denote the inverse function as g(x) = f^(-1)(x), then the derivative of the inverse function at x = g^(-1)(y) is given by:

[g^(-1)'(y)] = 1 / [f'(x)]

With the value of x that we found in Step 2, we can now calculate the derivative of the inverse function at y = -15.

Please note that without solving the equation or obtaining an approximation for the value of x, we cannot provide the exact derivative of the inverse function at y = -15.