Questions: Find the derivative of the function. h(x)=(x^8-2)^3

Solution Steps

To find the derivative of the function \( h(x) = (x^8 - 2)^3 \), we will use the chain rule. The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Here, let \( u = x^8 - 2 \), so \( h(x) = u^3 \). First, find the derivative of \( u^3 \) with respect to \( u \), and then multiply it by the derivative of \( u \) with respect to \( x \).

The function given is \( h(x) = (x^8 - 2)^3 \). To find the derivative, we apply the chain rule. Let \( u = x^8 - 2 \), so \( h(x) = u^3 \).

Differentiate \( u^3 \) with respect to \( u \). The derivative is: \[ \frac{d}{du}(u^3) = 3u^2 \]

Differentiate \( u = x^8 - 2 \) with respect to \( x \). The derivative is: \[ \frac{d}{dx}(x^8 - 2) = 8x^7 \]

Using the chain rule, the derivative of \( h(x) \) is: \[ h'(x) = \frac{d}{du}(u^3) \cdot \frac{d}{dx}(x^8 - 2) = 3u^2 \cdot 8x^7 \]

Substitute back \( u = x^8 - 2 \): \[ h'(x) = 3(x^8 - 2)^2 \cdot 8x^7 = 24x^7(x^8 - 2)^2 \]

Final Answer