High School

Find the derivative of the function:

\[ y = (7x^4 - 4x^2 + 4)^4 \]

Which of the following options represents the correct derivative of the function?

A) \( 28x^3 - 16x \)
B) \( 28x^3 + 16x \)
C) \( 28x^3 - 8x \)
D) \( 28x^3 + 8x \)

Answer :

Final answer:

The derivative of the function y = (7x^4 - 4x^2 + 4)^4 using the chain rule is 4*(7x^4 - 4x^2 + 4)^3 * (28x^3 - 8x), which none of the provided options correctly represent.

Explanation:

The question asks us to find the derivative of the function y = (7x^4 - 4x^2 + 4)^4. To solve this, we will use the chain rule of calculus, which is a technique for differentiating compositions of functions. We first recognize our outer function as u^4 where u = 7x^4 - 4x^2 + 4, and then we differentiate the inner function u with respect to x. The derivative of u is 28x^3 - 8x.

Applying the chain rule, we multiply the derivative of the outer function with respect to u, which is 4u^3, by the derivative of u with respect to x, giving us 4*(7x^4 - 4x^2 + 4)^3 * (28x^3 - 8x). None of the provided options correctly represents the derivative of the function because they are not in the form that applies the chain rule properly to this function.