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Chapter 6: Problem 17

If \(y=\log (\sin (3 x+5))\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
The derivative of the function is \(\frac{d y}{d x} = \frac{1}{\sin(3x+5)} \cdot 3\cos(3x+5)\)

Step by step solution

01

Identify the inner function and outer function

Let's denote the inner function as \(u = \sin(3x+5)\) and the outer function as \(v = \log(u)\).
02

Differentiate the outer function

The derivative of the outer function \(v=\log(u)\) with respect to \(u\) is \(v'=1/u\).
03

Differentiate the inner function

The derivative of the inner function \(u=\sin(3x+5)\) with respect to \(x\) is \(3\cos(3x+5)\). This is because the derivative of \(\sin(x)\) is \(\cos(x)\) and the constant \(3x+5\) is multiplied due to the chain rule.
04

Apply the chain rule to calculate the final derivative

The chain rule states that \(dy/dx = v'\cdot u'\). Substitute \(v'\) and \(u'\) with the results from step 2 and step 3 respectively. The final derivative is therefore \(dy/dx = \frac{1}{\sin(3x+5)} \cdot 3\cos(3x+5)\).

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