Question

Find the derivative of the function. \(h(x)=\sec x^{3}\)

Short answer

The derivative of the function \(h(x)=\sec x^{3}\) is \((\sec x^{3} \cdot \tan x^{3}) \cdot 3x^{2}\).

Step-by-step solution

Identify the Inner and Outer Functions

The given function \(h(x)=\sec x^{3}\) is a composition of two functions: the outer function is \(\sec u\) and the inner function is \(u=x^{3}\).

Compute the Derivatives

The derivative of the secant function is \(\sec u \cdot \tan u\), and the derivative of \(x^{3}\) is \(3x^{2}\). You will use these derivatives in the next step.

Apply the Chain Rule

According to the chain rule, to compute the derivative of the composite function, we first differentiate the outer function, then multiply it by the derivative of the inner function. Therefore, the derivative of our function is \((\sec u \cdot \tan u) \cdot 3x^{2}\) where \( u = x^{3}\). Replace \(u\) with \(x^3\) in the equation to find the final answer.

Final Answer

After replacing \(u\) in the equation we then have \((\sec x^{3} \cdot \tan x^{3}) \cdot 3x^{2}\)