Differentiate the following from the first principlex - Vidyadip

Question

Differentiate the following from the first principle$\text{x}\sin\text{x}$

✓

Answer

We have,$\text{f}(\text{x})=\text{x}\sin\text{x}$
$\because\text{f}'(\text{x})=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{(\text{x}+\text{h})\sin(\text{x}+\text{h})-\text{x}\sin\text{x}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{x}(\sin(\text{x}+\text{h})-\sin\text{x})}{\text{h}}+\sin(\text{x}+\text{h})\ $ $\Big[\sin\text{c}-\sin\text{d}=2\cos\frac{\text{c}+\text{d}}{2}\sin\frac{\text{c}-\text{d}}{2}\Big]$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{x}\times2\cos\Big(\text{x}+\frac{\text{h}}{2}\Big)\sin\frac{\text{h}}{2}}{\text{h}}+\sin(\text{x}+\text{h})\ $ $\Big[\because\lim_\limits{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=\text{2x}\times\cos\text{x}\times\frac{1}{2}+\sin\text{x}$
$=\text{x}\times\cos\text{x}+\sin\text{x}$
$=\sin\text{x}+\text{x}\cos\text{x}$

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

Show that the straight lines $L_1 = (b + c)x + ay + 1 = 0, L_2 = (c + a)x + by + 1 = 0$ and $L_3 = (a + b)x + cy + 1 = 0$ are concurrent.

Prove that $\frac{1}{2}\tan\Big(\frac{\text{x}}{2}\Big)+\frac{1}{4}\tan\Big(\frac{\text{x}}{4}\Big)+...+\frac{1}{2^{\text{n}}}\tan\Big(\frac{\text{x}}{2^{\text{n}}}\Big)=\frac{1}{2^{\text{n}}}\cot\Big(\frac{\text{x}}{2^{\text{n}}}\Big)-\cot\text{x}$ for all $\text{n}\in\text{N}$ and $0<\text{x}<\frac{\pi}{2}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$

Sum the following series to n terms:
$\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ .....$

$4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)=(\text{a + b + c})^2$

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
$\text{A}\times(\text{B}\cup\text{C})=(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$

The variance of 20 observation is 5. If each observation is multiplied by 2, find the variance of the resulting observation.

The vertices of a quadrilateral are A (-2, 6), B (1, 2), C (10, 4) and D (7, 8). Find the equation of its diagonals.

Find the middle terms(s) in the expansion of:
$\Big(\frac{\text{p}}{\text{x}}+\frac{\text{x}}{\text{p}}\Big)^{9}$

Solve the following systems of linear inequations graphically:
$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{y}\geq1,\text{x}\geq0,\text{y}\geq0$