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Differentiate w.r.t. \(x\) the function in \((5x)^{\large 3 \cos 2x} \)

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  • $\large\frac{d}{dx}$$(\log y)=\large\frac{1}{y}$
  • $\large\frac{d}{dx}$$(\cos x)=-\sin x$
Step 1:
Let $y=(5x)^{\large 3\cos 2x}$
Taking $\log$ on both sides
$\log y=3\cos 2x\log 5x$
Step 2:
Differentiating with respect to $x$
$\large\frac{1}{y}\frac{dy}{dx}$$=-3.2\sin 2x\log 5x+3\cos 2x.\large\frac{1}{x}$
$\qquad=3\begin{bmatrix}\large\frac{\cos 2x}{x}\normalsize-2\sin 2x\log 5x\end{bmatrix}$
$\large\frac{dy}{dx}=$$y.3\begin{bmatrix}\large\frac{\cos 2x}{x}\normalsize-2\sin 2x\log 5x\end{bmatrix}$
$\quad\;=(5x)^{\large 3\cos 2x}.3\begin{bmatrix}\large\frac{\cos 2x}{x}\normalsize-2\sin 2x\log 5x\end{bmatrix}$
$\quad\;=(5x)^{\large 3\cos 2x}\begin{bmatrix}\large\frac{3\cos 2x}{x}\normalsize-6\sin 2x\log 5x\end{bmatrix}$
answered May 14, 2013 by sreemathi.v

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