If 3 and y= ^ -1 ( ), find dy dx . - Vidyadip

Question

If $\pi\leq\text{x}\leq3\pi$ and $\text{y}=\cos^{-1}(\cos\text{x}),$ find $\frac{\text{dy}}{\text{dx}}.$

✓

Answer

We have,
$\text{y}=\cos^{-1}(\cos\text{x})$
$\Rightarrow\text{y}=2\pi-\text{x}$
$\big[\because\cos^{-1}(\cos\text{x})=2\pi-\text{x},\text{if x}\in[\pi,2{\pi}]\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(2\pi-\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=0-1$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-1$

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