Question

exercises 3.6 the chain ru
score: 8.83/19 answered: 11/19
question 12
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let ( f ( x ) = - 4 sec ( 7 x ) )
( f ^ { prime } ( x ) = )
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Explanation:

Step1: Apply the constant multiple rule

If \(y = - 4\sec(7x)\), by the rule \((cf(x))^\prime=c\cdot f^\prime(x)\) (\(c=-4\)), we first find the derivative of \(\sec(u)\) where \(u = 7x\). The derivative of \(\sec(u)\) with respect to \(u\) is \(\sec(u)\tan(u)\).

Step2: Apply the chain rule

By the chain rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Here \(y=-4\sec(u)\) (\(u = 7x\)), \(\frac{dy}{du}=-4\sec(u)\tan(u)\) and \(\frac{du}{dx}=7\).

Step3: Substitute \(u = 7x\) back

Substitute \(u = 7x\) into \(\frac{dy}{dx}=-4\sec(u)\tan(u)\cdot7\). We get \(f^\prime(x)=-28\sec(7x)\tan(7x)\)

Answer:

\(-28\sec(7x)\tan(7x)\)