Limit Continuity And Differentiability Ques 30
- Given, $y=\frac{5 x}{3 \sqrt{(1-x)^2}}+\cos ^2(2 x+1)$, find $\frac{d y}{d x}$.
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Answer:
Correct Answer: 30.$\left\{\begin{array}{l}\frac{5}{3(1-x)^2}-2 \sin (4 x+2), x<1 \\ \frac{-5}{3(x-1)^2}-2 \sin (4 x+2), x>1\end{array}\right.$
Solution: Given, $y=\frac{5 x}{3|1-x|}+\cos ^2(2 x+1)$
$\Rightarrow \quad y= \begin{cases}\frac{5 x}{3(1-x)}+\cos ^2(2 x+1), & x<1 \\ \frac{5 x}{3(x-1)}+\cos ^2(2 x+1), & x>1\end{cases}$
The function is not defined at $x=1$.
$\Rightarrow \quad \frac{d y}{d x}= \begin{cases}\frac{5}{3}\left\{\frac{(1-x)-x(-1)}{(1-x)^2}\right\}-2 \sin (4 x+2), & x<1 \\ \frac{5}{3}\left\{\frac{(x-1)-x(1)}{(x-1)^2}\right\}-2 \sin (4 x+2), & x>1\end{cases}$
$\Rightarrow \quad \frac{d y}{d x}= \begin{cases}\frac{5}{3(1-x)^2}-2 \sin (4 x+2), & x<1 \\ -\frac{5}{3(x-1)^2}-2 \sin (4 x+2), & x>1\end{cases}$
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