Chapter 2: Q. 65 (page 234)

Use logarithmic differentiation to find the derivative of given function
$f (x) = (\sin x)^{\cos x}$

Short Answer

Expert verified

The derivative of given function is

$f^{'} (x) = (- \sin x \ln (\sin x) + \frac{\cos^{2} x}{\sin x}) (\sin x)^{\cos x}$

Step by step solution

Step 1.Given data

We have been given a function to find the derivative for them

$f (x) = (\sin x)^{\cos x}$

Step 2.Taking log on both side to get the differentiation

$\begin{aligned} f (x) = (\sin x)^{\cos x} \\ \ln f (x) = \ln (\sin x)^{\cos x} [Taking log in both sides] \\ \ln f (x) = \cos x \ln (\sin x) \\ \frac{f^{'} (x)}{f (x)} = \frac{d}{d x} (\cos x \ln (\sin x)) [\begin{matrix} Differentiate both sides \\ with respect to x \end{matrix}] \\ \frac{f^{'} (x)}{f (x)} = (\frac{d}{d x} (\cos x)) \ln (\sin x) + \cos x (\frac{d}{d x} \ln (\sin x)) \end{aligned}$

$\begin{aligned} = - \sin x \ln (\sin x) + \frac{\cos x}{\sin x} (\frac{d}{d x} \sin x) \\ = - \sin x \ln (\sin x) + \frac{\cos x}{\sin x} (\cos x) \end{aligned}$

Using product rule and chain rule

$f^{'} (x) = (- \sin x \ln (\sin x) + \frac{\cos x}{\sin x} (\cos x)) f (x)$

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Use logarithmic differentiation to find the derivative of given function
$f (x) = (\sin x)^{\cos x}$

Short Answer

Step by step solution

Step 1.Given data

Step 2.Taking log on both side to get the differentiation

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Use logarithmic differentiation to find the derivative of given functionf(x)=(sin⁡x)cos⁡x

Short Answer

Step by step solution

Step 1.Given data

Step 2.Taking log on both side to get the differentiation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Use logarithmic differentiation to find the derivative of given function
$f (x) = (\sin x)^{\cos x}$