Differentiate the following functions with respect to x: (x+ x^2+… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\log(\text{x}+\sqrt{\text{x}^2+1})$

✓

Answer

Let $\text{y}=\log(\text{x}+\sqrt{\text{x}^2+1})$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dy}}\log\big(\text{x}+\sqrt{\text{x}^2+1}\big)$
$=\frac{1}{\text{x}+\sqrt{\text{x}^2+1}}\frac{\text{d}}{\text{dx}}\Big(\text{x}+\big(\text{x}^2+1\big)^\frac{1}{2}\Big)$
[Using chain rule]
$=\frac{1}{\text{x}+\sqrt{\text{x}^2+1}}\Big[1+\frac{1}{2}\big(\text{x}^2+1\big)^{\frac{1}{2}-1}\frac{\text{d}}{\text{dx}}\big(\text{x}^2+1\big)\Big]$
$=\frac{1}{\text{x}+\sqrt{\text{x}^2+1}}\Big[1+\frac{1}{2\sqrt{\text{x}^2+1}}\times2\text{x}\Big]$
$=\frac{1}{\text{x}+\sqrt{\text{x}^2+1}}\Big[\frac{\sqrt{\text{x}^2+1}+\text{x}}{\sqrt{\text{x}^2+1}}\Big]$
$=\frac{1}{\sqrt{\text{x}^2+1}}$
So,
$\frac{\text{d}}{\text{dx}}\Big(\log\big(\text{x}+\sqrt{\text{x}^2+1}\big)\Big)=\frac{1}{\sqrt{\text{x}^2+1}}$

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 "5 Marks Questions" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b - 4. Prove that * is neither commutative nor associative on Z.

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from,
LONDON.

If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\sqrt{\frac{\text{y}^2}{\text{x}^2}-1}$

Five defective mangoes are acciedently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_{0}\frac{\sin\text{x}\cos\text{x}}{\cos^2\text{x}+3\cos\text{x}+2}\text{ dx}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{if }\text{ x}<0\\2\text{x}+3,&\text{ x}\geq0\end{cases}$

Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}$

Evaluate the following integrals as limit of sum:
$\int\limits^2_0(\text{x}+3)\text{dx}$

Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height h and radius r is one-third of the height of the cone, and the greatest volume of the cylinder is $\frac4 9 $times the volume of the cone.