Logarithmic differentiation Use logarithmic differentiationevaluate f'1x2.totanº xf1x215x +326

To evaluate the function, fx=tan10x15x+326 using logarithmic differentiation. Solution: The give…

Answered: Logarithmic differentiation Use…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Logarithmic differentiation Use logarithmic differentiation
evaluate f'1x2.
to
tanº x
f1x2
15x +
326

Expert Solution

Step 1

To evaluate the function, $f (x) = \frac{\tan^{10} x}{15 x + 32^{6}}$ using logarithmic differentiation.

Solution:

The give function is $f (x) = \frac{\tan^{10} x}{15 x + 32^{6}}$

Take log both the sides we get,

$\log [f (x)] = \log (\frac{\tan^{10} x}{15 x + 32^{6}})$

Now differentiation the above function with respect to $x$ ,

Step 2

$\begin{array}{rcl} \frac{1}{f (x)} \frac{d [f (x)]}{d x} & = & (\frac{15 x + 32^{6}}{\tan^{10} x}) \times [10 \tan^{9} x s e c^{2} x {(15 x + 32^{6})}^{- 1} - 15 {(15 x + 32^{6})}^{- 2} \tan^{10} x] \\ = & (\frac{15 x + 32^{6}}{\tan^{10} x}) [\frac{10 \tan^{9} x s e c^{2} x}{(15 x + 32^{6})} - \frac{15 \tan^{10} x}{{(15 x + 32^{6})}^{2}}] \\ = & [\frac{10 \tan^{9} x s e c^{2} x (15 x + 32^{6}) - 15 \tan^{10} x}{\tan^{10} x (15 x + 32^{6})}] \end{array}$

Simplifying the above expression as,

$\begin{array}{rcl} \frac{1}{f (x)} \frac{d [f (x)]}{d x} & = & [\frac{10 s e c^{2} x (15 x + 32^{6}) - 15 \tan x}{\tan x (15 x + 32^{6})}] \\ \frac{d [f (x)]}{d x} & = & f (x) [\frac{10 s e c^{2} x (15 x + 32^{6}) - 15 \tan x}{\tan x (15 x + 32^{6})}] \end{array}$