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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

User Orlyyn
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1 Answer

6 votes

Final answer:

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).

User Weeraa
by
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