Questions: Find the derivative, f'(x), of the function f(x)=(5 x^4-2 x^2-3)^4 A. f'(x)=4(5 x^4-2 x^2-3)^3(20 x^3-4 x) B. f'(x)=4(5 x^4-2 x^2-3)^5(20 x^3-4 x) C. f'(x)=4(20 x^3-4 x)^3(60 x^2-4) D. f'(x)=4(20 x^3-4 x)^3 E. f'(x)=4(20 x^3-4 x)^5

Solution Steps

Let \( f(x) = \left(5x^4 - 2x^2 - 3\right)^4 \). We will differentiate this function using the chain rule.

Define the inner function as \( u = 5x^4 - 2x^2 - 3 \). Then, the outer function can be expressed as \( f(u) = u^4 \).

Using the chain rule, the derivative of the outer function \( f(u) = u^4 \) with respect to \( u \) is: \[ \frac{df}{du} = 4u^3 \]

Next, we differentiate the inner function \( u \) with respect to \( x \): \[ \frac{du}{dx} = \frac{d}{dx}(5x^4 - 2x^2 - 3) = 20x^3 - 4x \]

Now, we apply the chain rule to find the derivative \( f'(x) \): \[ f'(x) = \frac{df}{du} \cdot \frac{du}{dx} = 4u^3 \cdot (20x^3 - 4x) \]

Substituting \( u = 5x^4 - 2x^2 - 3 \) back into the expression, we get: \[ f'(x) = 4(5x^4 - 2x^2 - 3)^3 \cdot (20x^3 - 4x) \]

Final Answer