Differentiate the following functions with respect to x: x 1+ - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\text{x}\sin\text{x}}{1+\cos\text{x}}$

✓

Answer

We have, $\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}\sin\text{x}}{1+\cos\text{x}}\Big)$
Using Quotient rule, we get
$=\frac{(1+\cos\text{x})\frac{\text{d}}{\text{dx}}(\text{x}\sin\text{x})-(\text{x}\sin\text{x})\frac{\text{d}}{\text{dx}}(1+\cos\text{x})}{(1+\cos\text{x})^2}$
$=\frac{(1+\cos\text{x})\Big(\text{x}\frac{\text{d}}{\text{dx}}\sin\text{x}+\sin\text{x}\frac{\text{d}}{\text{dx}}\Big)-\text{x}\sin\text{x}(-\sin\text{x})}{(1+\cos\text{x})^2}$ [Used product rule]
$=\frac{(1+\cos\text{x})(\text{x}\cos\text{x}+\sin\text{x})+\text{x}\sin^2\text{x}}{(1+\cos\text{x})^2}$
$=\frac{\text{x}\cos\text{x}+\text{x}\cos^2\text{x}+\sin\text{x}+\sin\text{x}\cos\text{x}+\text{x}\sin^2\text{x}}{(1+\cos\text{x})^2}$
$=\frac{(\text{x}\cos\text{x}+\sin\text{x}+\sin\text{x}\cos\text{x})+\text{x}(\sin^2\text{x}+\cos^2\text{x})}{(1+\cos\text{x})^2}$
$=\frac{\sin\text{x}+\sin\text{x}\cos\text{x}+\text{x}(\cos\text{x}+\sin^2\text{x}+\cos^2\text{x})}{(1+\cos\text{x})^2}$
$=\frac{\sin\text{x}(1+\cos\text{x})+\text{x}(\cos\text{x}+1)}{(1+\cos\text{x})^2}$
$=\frac{(\text{x}+\sin\text{x})(\cos\text{x}+1)}{(1+\cos\text{x})^2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Derivatives→Derivatives — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

prove that:
$\sin(\text{B}-\text{C})\cos(\text{A}-\text{D})+\sin(\text{C}-\text{A})\cos(\text{B}-\text{D})+\sin(\text{A}-\text{B})\cos(\text{C}-\text{D})=0$

Differentiate the following from first principle: $\frac{1}{\sqrt{3-\text{x}}}$

Prove the following identities: $\frac{\sin^3+\cos^3\text{x}}{\sin\text{x}+\cos\text{x}}+\frac{\sin^3\text{x}-\cos^3\text{x}}{\sin\text{x}-\cos\text{x}}=2$

Following are the marks obtained$,$ out of $100, $by two students Ravi and Hashina in $10$ tests.

Ravi	$25$	$50$	$45$	$30$	$72$	$42$	$36$	$48$	$35$	$60$
Hashima	$10$	$70$	$50$	$20$	$95$	$55$	$42$	$60$	$48$	$80$

Who is more intelligent and who is more consistent?

One diameter of the circle circumscribing the rectangle ABCD is 4y = x + 7. If the coordinates of Aand B are (-3, 4) and (5, 4) respectively, find the equation of the circle.

$i$. If $f(x)=\left\{\begin{array}{cl}|x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0\end{array}\right.$, for what values $(s)$ of a does $\lim _{x \rightarrow a} f(x)$ exist?
$ii$. Find the derivative of the function $\cos \left(x-\frac{\pi}{8}\right)$ from the first principle.

An urn contains $7$ white, $5$ black and $3$ red balls. Two balls are drawn at random. Find the probability that

Both the balls are red.
One ball is red and the other is black.
One ball is white.

If $^{24}C_x =\ ^{24}C_{2x+3},$ Find x.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{8}}\frac{\cot4\text{x}-\cos4\text{x}}{(\pi-8\text{x})^3}$

Prove that $\sin\text{x}+\sin3\text{x}+...+\sin(\text{2n-1})\text{x}=\frac{\sin ^2\text{nx}}{\sin\text{x}}$ for all $\text{n}\in\text{N}.$