CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Differentiate the function w.r.t. $$x$$.
$$\displaystyle x^{x \cos x} + \frac{x^2 + 1}{x^2 - 1}$$


Solution

Let $$\displaystyle y = x^{x \cos x} + \frac{x^2 + 1}{x^2 - 1}$$
Also, let $$\displaystyle u = x^{x \cos x}$$ and $$\displaystyle v = \frac{x^2 + 1}{x^2 - 1}$$
$$\therefore y = u + v$$
$$\Rightarrow \displaystyle \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}$$             ...(1)
$$\displaystyle u = x^{x \cos x}$$
$$\Rightarrow \displaystyle \log u = \log (x^{x \cos x})$$
$$\Rightarrow \displaystyle \log u = x \cos x \log x$$
Differentiating both sides with respect to $$x$$, we obtain
$$\displaystyle
\frac{1}{u} \frac{du}{dx} = \frac{d}{dx} (x) . \cos x . \log x  + x
\frac{d}{dx} (\cos x) . \log x + x \cos x . \frac{d}{dx} (\log x)$$
$$\Rightarrow \displaystyle \frac{du}{dx} = u \left[ 1 . \cos x . \log x + x .  (- \sin x) \log x + x \cos x . \frac{1}{x} \right] $$
$$\Rightarrow \displaystyle \frac{du}{dx} = x^{x \cos x} \left( \cos x . \log x - x \sin x \log x + \cos x \right) $$
$$\Rightarrow \displaystyle
\frac{du}{dx} = x^{x \cos x} \left[ \cos x (1 + \log x) - x \sin x \log x
\right] $$                                    ...(2)
$$\displaystyle v = \frac{x^2 + 1}{x^2 - 1}$$
$$\Rightarrow \displaystyle \log v = \log {(x^2 + 1)} - \log {(x^2 - 1)}$$
Differentiating both sides with respect to $$x$$, we obtain
$$ \displaystyle \frac{1}{v} \frac{dv}{dx} = \frac{2x}{x^2 + 1} - \frac{2x}{x^2 - 1}$$
$$\Rightarrow \displaystyle \frac{dv}{dx} = v \left[ \frac{2x(x^2 - 1) - 2x(x^2 + 1)}{(x^2 + 1) (x^2 - 1)} \right] $$
$$\Rightarrow \displaystyle \frac{dv}{dx} = \frac{x^2 + 1}{x^2 - 1} \times \left[ \frac{- 4x }{(x^2 + 1) (x^2 - 1)} \right] $$
$$\Rightarrow \displaystyle \frac{dv}{dx} = \frac{- 4x }{(x^2 - 1)^2} $$                        ...(3)
From (1), (2) and (3), we obtain
$$\Rightarrow \displaystyle \frac{dy}{dx} = x^{x
\cos x} \left[ \cos x (1 + \log x) - x \sin x \log x \right]
-  \frac{4x }{(x^2 - 1)^2} $$

Mathematics
RS Agarwal
Standard XII

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
Same exercise questions
View More


similar_icon
People also searched for
View More



footer-image