Question

find dy/dx if y = sec(8x²).

Explanation:

Step1: Recall the chain rule and derivative of secant

The chain rule states that if \( y = f(g(x)) \), then \( \frac{dy}{dx}=f'(g(x))\cdot g'(x) \). The derivative of \( \sec(u) \) with respect to \( u \) is \( \sec(u)\tan(u) \), where \( u = 8x^{2} \).

Step2: Differentiate the outer function

Let \( u = 8x^{2} \), so \( y=\sec(u) \). Then \( \frac{dy}{du}=\sec(u)\tan(u) \).

Step3: Differentiate the inner function

Differentiate \( u = 8x^{2} \) with respect to \( x \). Using the power rule \( \frac{d}{dx}(ax^{n})=nax^{n - 1} \), we get \( \frac{du}{dx}=16x \).

Step4: Apply the chain rule

By the chain rule \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). Substitute \( \frac{dy}{du}=\sec(u)\tan(u) \) and \( \frac{du}{dx}=16x \), and then replace \( u \) with \( 8x^{2} \). So \( \frac{dy}{dx}=\sec(8x^{2})\tan(8x^{2})\cdot16x \).

Answer:

\( 16x\sec(8x^{2})\tan(8x^{2}) \)