Question

If f(x)=(x²−3)4, then f′(1)=
(A) −64
(B) −32
(C) −16
(D) 32

Answer 1

To find f'(1) for the function f(x)=(x²-3)⁴, the chain rule is used to differentiate the outer and inner functions and then plug in x=1. After computing, the result is -64. Thus, the correct answer is (A) -64.

Step-by-step explanation:

To find f'(1) for the function f(x)=(x²-3)⁴, we need to apply the chain rule for differentiation. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. In this case, our outer function is ‘u⁴’ where ‘u’ is our inner function ‘x² - 3’.

First, differentiate the outer function:
Outer derivative: If we have ‘u⁴’, its derivative is 4u³.

Next, differentiate the inner function:
Inner derivative: The derivative of ‘x² - 3’ is 2x

Now apply the chain rule:
f'(x) = 4(x² - 3)³ × 2x

To find f'(1), substitute ‘x’ with 1 in the derivative we found:
f'(1) = 4((1)² - 3)³ × 2×1 = 4(-2)³×2
= 4(-8)×2
= -64

Therefore, the correct answer is (A) −64.