Question

Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$

Short answer

The derivative of the function \(h(x) = (6x - x^3)^2\) is \(h'(x) = 12x - 2x^3 - 18x^3 + 6x^5\).

Step-by-step solution

Identify the Inner Function and Its Derivative

The inner function, \(u\), is the part of the function that is being raised to a power. In this function, \(u = 6x - x^3\). To find its derivative, \(u'\), apply the power rule separately to \(6x\) and \(-x^3\) to get \(u' = 6 - 3x^2\).

Apply the General Power Rule

The General Power Rule states that the derivative of \(u^n\) is \(nu^{n-1}u'\), where \(u'\) is the derivative of \(u\). In this function, \(n = 2\). Thus, applying the General Power Rule gives: \[ h'(x) = 2(6x - x^3)^{2-1}(6 - 3x^2) \].

Simplify the Expression

The expression can be simplified by performing the indicated operations: \[ h'(x) = 2(6x - x^3)(6 - 3x^2) \]. After multiplying out and combining like terms, the derivative of the function is found to be \( h'(x) = 12x - 2x^3 - 18x^3 + 6x^5\).