Question

h(x)=\sec ^{2}(4 x)
which sequence of rules can be used in order to differentiate h
in its current form?
choose 1 answer:
a chain rule, then product rule
b product rule, then product rule again
c product rule, then chain rule
d chain rule, then chain rule again

Explanation:

Step1: Analyze the function structure

The function \(h(x)=\sec^{2}(4x)\) can be seen as a composition of functions. Let \(u = 4x\), and \(y=\sec^{2}(u)\) (where \(y = v^{2}\) and \(v=\sec(u)\)).

Step2: Apply the chain - rule concept

The chain - rule states that if \(y = f(g(x))\), then \(y^\prime=f^\prime(g(x))\cdot g^\prime(x)\).
For \(y = v^{2}\) with \(v=\sec(u)\) and \(u = 4x\), first, when differentiating \(y\) with respect to \(v\) (\(\frac{dy}{dv}=2v\)), then differentiating \(v\) with respect to \(u\) (\(\frac{dv}{du}=\sec(u)\tan(u)\)), and finally differentiating \(u\) with respect to \(x\) (\(\frac{du}{dx}=4\)).
We use the chain - rule twice. First, for \(y = v^{2}\) (where \(v\) is a function of \(u\)) and then for \(v=\sec(u)\) (where \(u\) is a function of \(x\)).

Answer:

D. Chain rule, then chain rule again