Question

If d/dx(f(4x⁵)) = 8x², calculate f'(x).

a) f'(x) = 32x⁶
b) f'(x) = 40x⁴
c) f'(x) = 20x⁴
d) f'(x) = 10x²

Answer 1

Upon applying the chain rule to the given derivative
`d/dx(f(4x⁵)) = 8x²` and simplifying the expression, we find that
`f'(x) = 2 / (5x²)` . This result is not listed among the provided multiple-choice options, indicating that there may be an error in the question as presented.

Step-by-step explanation:

To calculate
`f'(x)` when given that the derivative
`d/dx(f(4x⁵)) = 8x²` , we need to apply the chain rule. We'll call the inner function
`g(x) = 4x⁵` , so our outer function is
`f(g(x))`. When applying the chain rule, we get:

`f'(g(x)) × g'(x) = 8x²`

First, let's find g'(x):

`g'(x) = d/dx(4x⁵) = 20x⁴`

Now, let's denote
`f'(g(x)) as h(x)`. The given derivative becomes:

`h(x) × 20x⁴ = 8x²`

To find
`h(x)`, we divide both sides by
`20x⁴` :

`h(x) = 8x² / (20x⁴)`

`h(x)` simplifies to:

`h(x) = 8 / (20x²)h(x) = 2 / (5x²)`

Recall that
`h(x)` represents
`f'(g(x))`, and since
`g(x) = 4x⁵`, we replace
`g(x)` with
`x` to find
`f'(x)`:

`f'(x) = 2 / (5x²)`

The correct answer is not in the options provided since there seems to be an error.
`f'(x)` simplifies to
`2/(5x²)` and not to any of the options given (a, b, c, d). So,
`f'(x) = 2/(5x²)` is the correct derivative function of
`f(x)`.