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  4. If r=[2 φ+ cos 2(2 φ+π / 4)]1 / 2, then what is the value of the derivative of d r / d φ at φ=π / 4 ?

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Q. If $r=\left[2 \phi+\cos ^{2}(2 \phi+\pi / 4)\right]^{1 / 2}$, then what is the value of the derivative of $d r / d \phi$ at $\phi=\pi / 4 ?$

BITSATBITSAT 2005

Solution:

Given $r=\left[2 \phi+\cos ^{2}(2 \phi+\pi / 4)\right]^{1 / 2}$
$\operatorname{dr} / d \phi=(1 / 2)\left[2 \phi+\cos ^{2}(2 \phi+\pi / 4)\right]^{-1 / 2}[2-2 \sin (4 \phi+\pi / 2)]$
$=\left[2 \phi+\cos ^{2}(2 \phi+\pi / 4)\right]^{-1 / 2}[1-\cos (4 \phi)]$
dr/d\phiat $\phi=\pi / 4$
$=\left[2 \pi / 4+\cos ^{2}(2 \pi / 4+\pi / 4)\right]^{-1 / 2}[1-\cos (4 \pi / 4)]$
$=\left[\pi / 2+\cos ^{2}(\pi / 2+\pi / 4)\right]^{-1 / 2}[1-\cos (\pi)]$
$=\left[\pi / 2+\cos ^{2}(3 \pi / 4)\right]^{-1 / 2}[1+1]$
$=\left[\pi / 2+\cos ^{2}(3 \pi / 4)\right]^{-1 / 2} \times 2$
$=[\pi / 2+1 / 2]^{-1 / 2} \times 2$
$=2[(\pi+1) / 2]^{-1 / 2}$
$=2 \sqrt{2} / \sqrt{(} \pi+1)$
$=2(2 / \pi+1)^{1 / 2}$