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Differentiate w.r.t. \(x\) the function in \( \cos \: (a \cos x + b \sin x) \) , for some constant \(a\) and \(b\)

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  • $\large\frac{dy}{dx}=\large\frac{dy}{dt}$$\times\large\frac{dt}{dx}$
Step 1:
Put $a\cos x+b\sin x=t$
$y=\cos t$
$t=a\cos x+b\sin x$
Differentiating with respect to $t$
$\large\frac{dt}{dx}$$=-a\sin x+b\cos x$
Step 2:
$\large\frac{dy}{dx}=\large\frac{dy}{dt}$$\times\large\frac{dt}{dx}$
$\quad\;=-\sin t.-a\sin x+b\cos x$
By substituting the value of $t$
$\quad\;=-\sin(a\cos x+b\sin x)(-a\sin x+b\cos x)$
$\quad\;=(a\sin x-b\cos x)\sin(a\cos x+b\sin x)$
answered May 14, 2013 by sreemathi.v
 

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