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\frac{d}{d x} e^{x \cos x} is equal to ?

Option: 1

e^{x \cos x}(\cos x+x \sin x)


Option: 2

e^{x \cos x}(\sin x-x \cos x)


Option: 3

e^{x \cos x}(\sin x+x \cos x)


Option: 4

e^{x \cos x}(\cos x-x \sin x)


Answers (1)

best_answer

 

 

Differentiation Using Logarithm -

Differentiation Using Logarithm

Till now what we have studied using that, we can take derivatives of functions of the form y = (g(x))n. for certain values of n, as well as functions of the form y = bg(x) , where b > 0 and b ≠ 1. Unfortunately, we still do not know the derivatives of functions such as y = x x or y = x π . These functions require a technique called logarithmic differentiation, which allows us to differentiate any function of the form h(x) = g(x) f(x) .

1.

    \\\mathrm{Let,\;\;\;\;\;y=\left ( f(x) \right )^{g(x)}}\\\mathrm{Take\;\log\;both\;side}\\\mathrm{\;\;\;\;\;\;\;\log y=g(x)\log f(x)}\\\mathrm{Differentiate\;with\;respect\;to\;x}\\\mathrm{\;\;\;\;\;\;\;\frac{1}{y}\cdot\frac{dy}{dx}=g(x)\cdot \frac{1}{f(x)}\cdot\frac{d}{dx}\left ( f(x) \right )+\log f(x)\cdot \frac{d}{dx}\left ( g(x) \right )}\\\\\mathrm{\therefore\;\;\;\;\;\;\;\;\;\frac{dy}{dx}=y \left [ \frac{g(x)}{f(x)}\cdot\frac{d}{dx}\left ( f(x) \right )+\log f(x)\cdot \frac{d}{dx}\left ( g(x) \right ) \right ]}\\\\\mathrm{or\;\;\;\;\;\;\;\;\;\mathbf{\frac{dy}{dx}=\left ( f(x) \right )^{g(x)} \left [ \frac{g(x)}{f(x)}\cdot\frac{d}{dx}\left ( f(x) \right )+\log f(x)\cdot \frac{d}{dx}\left ( g(x) \right ) \right ]}} 
     
Note:

Some more methods to differentiate \mathrm{y}=[f(x)]^{g(x)}

\\\text { (i) } \;\;\;\;\;\;y=[f(x)]^{g(x)}=e^{g(x) \log _{e} f(x)} \\\\ {\quad\quad \therefore \;y^{\prime}=e^{g(x) \log _{e} f(x)} \times\left[g^{\prime}(x) \log _{e} f(x)+g(x) \frac{f^{\prime}(x)}{f(x)}\right]}.
 

\\\text { (ii) } \;\;\;\;\;\;\;y=[f(x)]^{g(x)} \\\\ {\quad\ \therefore \;\;\frac{d y}{d x}=\text { Differential of } y \text { treating } f(x) \text { as constant }}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\text { Differential of } y \text { treating } g(x) \text { as constant. }}

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\text{Let }y=e^{x \cos x}\\ \log y=x \cos x \\ \frac{1}{y} \frac{d y}{d x}=\cos x-x \sin x\\ \frac{d y}{d x}=e^{x \cos x}(\cos x-x \sin x)

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seema garhwal

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