Differentiate the following from the first principle (3x+1) - Vidyadip

Question

Differentiate the following from the first principle$\sqrt{\sin(\text{3x}+1)}$

✓

Answer

We have, $\text{f}(\text{x})=\sqrt{\sin(\text{3x}+1)}$
$\because\text{f}'(\text{x})=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\sqrt{\sin3(\text{x}+\text{h})+1}-\sqrt{\sin(\text{3x}+1)}}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\sqrt{\sin(\text{3x}+\text{3h})+1}-\sqrt{\sin(\text{3x}+1)}}{\text{h}}\times\frac{{\sqrt{\sin(\text{3x}+\text{3h})+1}+\sqrt{\sin(\text{3x}+1)}}}{{\sqrt{\sin(\text{3x}+\text{3h})+1}+\sqrt{\sin(\text{3x}+1)}}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\sin(\text{3x}+\text{3h}+1)-\sin(\text{3x}+1)}{\text{h}\big(\sqrt{\sin(\text{3x}+\text{3h})+1}+\sqrt{\sin(\text{3x}+1)}\big)}$
$=\lim_\limits{\text{h}\rightarrow0}2\cos\Big(\text{3x}+1+\frac{\text{3h}}{2}\Big)\times\frac{\sin\frac{\text{3h}}{2}}{\frac{\text{3h}}{2}}\times\frac{3}{2}\times\frac{1}{\sqrt{\sin(\text{3x}+\text{3h})+1}+\sqrt{\sin(\text{3x}+1)}}$
$=\frac{3\cos(\text{3x}+1)}{2\sqrt{\sin(\text{3x}+1)}}$ $\Bigg[\lim_\limits{\text{h}\rightarrow0}\frac{\sin\frac{\text{3h}}{2}}{\frac{\text{3h}}{2}}=1\Bigg]$

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

Differentiate the following functions by the product rule and the other method and verify that the answer from both the methods is the same.$(3\sec\text{x}-4\text{cosec}\text{x})(-2\sin\text{x}+5\cos\text{x})$

A railway train is travelling on a circular curve of 1500 metres radius at the rate of 66km/ hr. Through what angle has it turned in 10 seconds?

Find the equation of the hyperbola whose focus is (1, 3), directrix is x + y - 1 = 0 and eccentricity = 2

In an examination, a student has to answer 4 questions out of 5 questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.

If $\text{A}\times\text{b}\subseteq\text{C}\times\text{D and A}\times\text{B}=\phi,$ prove that $\text{A}\subseteq\text{C and B}\subseteq\text{D}$

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}.$ Find them.

Prove the following by the principle of mathematical induction: $1 + 3 + 3^2 + ... + 3^{\text{n}-1}=\frac{3^\text{n}-1}{2}$

The base of an equilateral triangle with side 2a lies along the they-axis such that the mid-point of the base is at the origin. Find the vertices of the triangle.

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

Ravi:	25	50	45	30	70	42	36	48	35	60
Hashina:	10	70	50	20	95	55	42	60	48	80

Who is more intelligent and who is more consistent?

Find the equation of the straight line drawn through the point of intersection of the lines x + y = 4 and 2x - 3y = 1 and perpendicular to the line cutting off intercepts 5, 6 on the axes.