Question

exercises 3.6 the chain
score: 5/19 answered: 5/19
question 6
textbook videos +
let ( f(x)=5 csc (4 x) )
( f^{prime}(x)= )
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Explanation:

Step1: Recall the chain rule

If \(y = f(g(x))\), then \(y^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Also, the derivative of \(\csc(u)\) with respect to \(u\) is \(-\csc(u)\cot(u)\). Let \(u = 4x\), so \(f(x)=5\csc(u)\)

Step2: Find the derivative of the outer - function

The derivative of \(y = 5\csc(u)\) with respect to \(u\) is \(y^\prime_u=- 5\csc(u)\cot(u)\)

Step3: Find the derivative of the inner - function

The derivative of \(u = 4x\) with respect to \(x\) is \(u^\prime_x = 4\)

Step4: Apply the chain rule

By the chain rule \(f^\prime(x)=\frac{df}{du}\cdot\frac{du}{dx}\). Substitute \(u = 4x\), \(\frac{df}{du}=-5\csc(u)\cot(u)\) and \(\frac{du}{dx}=4\) into the formula. We get \(f^\prime(x)=-5\csc(4x)\cot(4x)\cdot4=-20\csc(4x)\cot(4x)\)

Answer:

$-20\csc(4x)\cot(4x)$