Answered: 41. y = 23+ sin x

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Analytic Trigonometry

Section2.3: Solving Trigonometric Equations

Problem 9ECP: Solve 4tan2x+5tanx6=0.

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Question

Derivatives Find the derivative of the following functions

Expert Solution

Step 1

Consider the provided function,

$y = 2^{3 + \sin x} . . . . . . (1)$

Find the derivative of the above functions.

First, we taking log both the sides.

$\ln y = \ln (2^{3 + \sin x})$

Apply the log property $\ln (m^{n}) = n \ln (m)$ .

So,

$\begin{array}{rcl} \ln y & = & \ln (2^{3 + \sin x}) \\ \ln y & = & (3 + \sin x) \ln (2) \end{array}$

Step 2

Now, the above equation differentiate with respect to x.

Apply the common derivative rule $\frac{d}{d x} (\ln x) = \frac{1}{x} \cdot 1 and \frac{d}{d x} (\sin x) = \cos x$ .

$\begin{array}{rcl} \frac{d}{d x} (\ln y) & = & \frac{d}{d x} [(3 + \sin x) \ln (2)] \\ \frac{1}{y} \frac{d y}{d x} & = & \ln (2) \frac{d}{d x} [(3 + \sin x)] \\ \frac{1}{y} \frac{d y}{d x} & = & \ln (2) \cdot (0 + \cos x) \\ \frac{1}{y} \frac{d y}{d x} & = & \ln (2) \cdot \cos x \\ \frac{d y}{d x} & = & y \cdot \ln (2) \cdot \cos x \end{array}$

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