Questions: Differentiate the following: f(x) = sec^(-1)(7x^3)

Solution Steps

To differentiate the function \( f(x) = \sec^{-1}(7x^3) \), we will use the chain rule and the derivative of the inverse secant function. The derivative of \( \sec^{-1}(u) \) with respect to \( u \) is \( \frac{1}{|u|\sqrt{u^2 - 1}} \). We will also need to differentiate the inner function \( 7x^3 \).

We start with the function \( f(x) = \sec^{-1}(7x^3) \). To differentiate this function, we will apply the chain rule and the derivative of the inverse secant function.

Using the derivative of the inverse secant function, we have: \[ f'(x) = \frac{1}{|u|\sqrt{u^2 - 1}} \cdot \frac{du}{dx} \] where \( u = 7x^3 \) and \( \frac{du}{dx} = 21x^2 \). Thus, we can express the derivative as: \[ f'(x) = \frac{21x^2}{|7x^3|\sqrt{(7x^3)^2 - 1}} = \frac{21x^2}{7|x^3|\sqrt{49x^6 - 1}} = \frac{3}{|x|^4\sqrt{49x^6 - 1}} \]

Final Answer