Chapter 2: Q 63. (page 234)

Use logarithmic differentiation to find derivative
$(\sin x)^{x}$

Short Answer

Expert verified

The derivative of given function is

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

Step by step solution

Step 1. Given information

We have been given

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

to find the derivative of this function .

Step 2.Finding derivative

Product rule for differentiation

(f g)^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)

Quotient rule for differentiation

${(\frac{f}{g})}^{'} (x) = \frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{(g (x))^{2}}$

Now ,

$\begin{aligned} \frac{d}{d x} (\sinh x) = \cosh x \\ \frac{d}{d x} (\cosh x) = \sinh x \\ \frac{d}{d x} (\tanh x) = {sech}^{2} x \\ \frac{d}{d x} (\sinh^{- 1} x) = \frac{1}{\sqrt{x^{2} + 1}} \\ \frac{d}{d x} (\cosh^{- 1} x) = \frac{1}{\sqrt{x^{2} - 1}} \\ \frac{d}{d x} (\tanh^{- 1} x) = \frac{1}{1 - x^{2}} \end{aligned}$

Step 3.Using log to get derivative

$\begin{aligned} f (x) = (\sin x)^{x} \\ \ln f (x) = \ln (\sin x)^{x} [Taking log in both sides] \\ \ln f (x) = x \ln (\sin x) \\ \frac{f^{'} (x)}{f (x)} = \frac{d}{d x} (x \ln (\sin x)) [\begin{matrix} Differentiate both sides \\ with respect to x \end{matrix}] \\ \frac{f^{'} (x)}{f (x)} = (\frac{d}{d x} (x)) \ln (\sin x) + x (\frac{d}{d x} \ln (\sin x)) \\ = \ln (\sin x) + \frac{x}{\sin x} (\frac{d}{d x} \sin x) [\begin{matrix} Using product rule \\ Using chain rule \\ \frac{d}{d x} \ln x = \frac{1}{x} \end{matrix}] \\ = \ln (\sin x) + \frac{x}{\sin x} (\cos x) [\frac{d}{d x} \sin x = \cos x] \\ f^{'} (x) = (\ln (\sin x) + \frac{x \cos x}{\sin x}) f (x) [Multiplying both sides by f (x)] \\ f^{'} (x) = (\ln (\sin x) + \frac{x \cos x}{\sin x}) (\sin x)^{x} \end{aligned}$

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Use logarithmic differentiation to find derivative
$(\sin x)^{x}$

Short Answer

Step by step solution

Step 1. Given information

Step 2.Finding derivative

Step 3.Using log to get derivative

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Use logarithmic differentiation to find derivative(sin⁡x)x

Short Answer

Step by step solution

Step 1. Given information

Step 2.Finding derivative

Step 3.Using log to get derivative

One App. One Place for Learning.

Most popular questions from this chapter

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Use logarithmic differentiation to find derivative
$(\sin x)^{x}$