Differentiate the following from first principle: 2x^2-1 - Vidyadip

Question

Differentiate the following from first principle:$\sqrt{2\text{x}^2-1}$

✓

Answer

We have, $\text{f(x)}=\sqrt{2\text{x}^2+1}$ $\because\text{f}'\text{(x)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{x+h})-\text{f}(\text{x})}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\sqrt{2(\text{x+h})^2+1}-\sqrt{2\text{x}^2+1}}{\text{h}}$ Multiplying Numerator and denominator by ${\sqrt{2(\text{x+h})^2+1}+\sqrt{2\text{x}^2+1}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\bigg\{2(\text{x+h})^2+1-(2\text{x}^2+1)\bigg\}}{\text{h}\bigg(\sqrt{2(\text{x+h})^2+1}+\sqrt{2\text{x}^2+1}\bigg)}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{2\text{x}^2+2\text{h}^2+4\text{xh}+1-2\text{x}^2-1}{\text{h}\bigg(\sqrt{2(\text{x+h})^2+1}+\sqrt{2\text{x}^2+1}\bigg)}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{4\text{xh}+2\text{h}^2}{\text{h}\sqrt{2(\text{x+h})^2+1}+\sqrt{2\text{x}^2+1}}$ $=\frac{4\text{x}}{2\sqrt{2\text{x}^2+1}}$ $=\frac{2\text{x}}{\sqrt{2\text{x}^2+1}}$ $$

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 "(Each question 4 marks)" from Derivatives→Derivatives — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If P(2n − 1, n) : P(2n + 1, n − 1) = 22 : 7 find n.

Find the mean deviation from the mean for following data:

Size	1	3	5	7	9	11	13	15
Frequency	3	3	4	14	7	4	3	4

If $^\text{n+5}\text{P}_\text{n+1}=-\frac{11(\text{n-1})}{2}\ ^\text{n+3}\text{P}_\text{n}$ find n.

Find the equation of the line, which passes through P (1, -7) and meets the axes at Aand B respectively so that 4 AP - 3 BP = 0.

For any two sets A and B, prove that $(\text{A}\cap\text{B})-\text{B = A}- \text{B}$

Calculate the standard deviation for the following data:

Class:	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequency:	9	17	43	82	81	44	24

Find the general solution for each of the following equations: $\cos4\text{x}=\cos2\text{x}$

Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. $\frac{\text{y}^2}{9}-\frac{\text{x}^2}{27}=1$

Prove that $\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}\cos\frac{16\pi}{15}=\frac{1}{16}$

Calculate the mean deviation about the median of the following observation: $38, 70, 48, 34, 63, 42, 55, 44, 53, 47$