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Differentiate the functions given in w.r.t. $x : $ $ ( \log \; x )^{\large cos \: x} $

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Toolbox:
  • $\log m^n=n\log m$
  • $\large\frac{d}{dx}$$\log x=\large\frac{1}{x}$
Step 1:
Let $y=(\log x)^{\large\cos x}$
Taking $\log$ on both sides
$\log y=\log(\log x)^{\large\cos x}$
Differentiating both sides with respect to $x$
$\large\frac{1}{y}\frac{dy}{dx}$$=(-\sin x)\log(\log x)+\cos x\large\frac{d}{dx}$$\log(\log x)$
$\qquad=-\sin x\log(\log x)+\cos x\begin{bmatrix}\large\frac{1}{\log x}\end{bmatrix}\large\frac{d}{dx}$$\log x$
$\qquad=-\sin x\log(\log x)+\large\frac{\cos x}{\log x}\large\frac{1}{x}$
$\qquad=-\sin x\log(\log x)+\large\frac{\cos x}{x\log x}$
Step 2:
Multiply by y
$\large\frac{dy}{dx}$$=y\begin{bmatrix}-\sin x \log(\log x)+\large\frac{\cos x}{x\log x}\end{bmatrix}$
$\quad=(\log x)^{\large\cos x}[-\sin x\ log(\log x)+\large\frac{\cos x}{x\log x}]$
answered May 8, 2013 by sreemathi.v
edited May 8, 2013 by sreemathi.v
 

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