If d/dx(f(3x^5))=9x^4, find f'(x)
I can't figure out what this means
The given equation, d/dx(f(3x^5)) = 9x^4, represents the derivative of the function f(3x^5) with respect to x. To find f'(x), we need to find the derivative of the function f(x).
To solve this problem, we can apply the chain rule of differentiation. The chain rule states that if we have a composite function f(g(x)), then its derivative can be found by multiplying the derivative of the outer function f'(g(x)) with the derivative of the inner function g'(x).
In this case, we have f(3x^5). Let's break it down:
- The outer function is f(x).
- The inner function is 3x^5.
We are given that d/dx(f(3x^5)) = 9x^4. This means that f'(3x^5) multiplied by the derivative of 3x^5 with respect to x (which is 15x^4) equals 9x^4. Therefore:
f'(3x^5) * 15x^4 = 9x^4
To find f'(3x^5), we can divide both sides of the equation by 15x^4:
f'(3x^5) = (9x^4)/(15x^4)
Simplifying, we get:
f'(3x^5) = 3/5
So, the derivative of f(x) is f'(x) = 3/5.