If d/dx(f(3x^5))=9x^4
Find f'(x)
nvm it's 3/5
say d/dx (g) = 9 x^4
then g = (9/5)x^5 + constant
so
f(3x^5) = (9/5) (3x^5)^5 + constant
f(x) = (9/5) x^5 + c
f'(x) = 9x
I mean 9 x^4
To find f'(x), we can make use of the chain rule. The chain rule states that if we have a composition of functions, say g(f(x)), then the derivative of this composition can be found by multiplying the derivative of the outer function (g') with the derivative of the inner function (f').
In this case, we have d/dx(f(3x^5)) = 9x^4. To find f'(x), we can use the chain rule to break down the given expression into smaller parts.
Let's denote u = 3x^5. So, f(3x^5) becomes f(u).
Using the chain rule, we have:
d/du(f(u)) * d/dx(3x^5) = 9x^4
Now, let's find the derivatives of f(u) and 3x^5:
d/du(f(u)) * d/dx(3x^5) = 9x^4
The right side is given as 9x^4. We can already see that the derivative of 3x^5 with respect to x is 9x^4. So, we can replace d/dx(3x^5) with 9x^4.
The equation becomes:
d/du(f(u)) * 9x^4 = 9x^4
Now, let's solve for d/du(f(u)):
d/du(f(u)) = (9x^4) / (9x^4)
The term (9x^4) cancels out on both sides of the equation, leaving us with:
d/du(f(u)) = 1
Since f(u) is a function of u, we can write this derivative as f'(u).
Therefore, we now have f'(u) = 1.
Finally, we substitute back the value of u, which is 3x^5:
f'(3x^5) = 1.
So, the derivative of f(x) is equal to 1.