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

Question

Differentiate the following functions with respect to x:
$\log\sqrt{\frac{\text{x}-1}{\text{x}+1}}$

✓

Answer

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

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 "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that $\text{y}=\text{A}\cos\text{x}+\text{B}\sin\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$

Find the points on the curve $y = x^3$ at which the slope of the tangent is equal to the $y-$coordinate of the point.

Evaluate: $\begin{vmatrix}\text{a}&\text{b}+\text{c}&\text{a}^2\\\text{b}&\text{c}+\text{a}&\text{b}^2\\\text{c}&\text{a}+\text{b}&\text{c}^2\end{vmatrix}$

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\Big(\sqrt{\tan\text{x}}+\sqrt{\cot}\text{x}\Big)\text{dx}$

A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by
$\text{M}=\frac{\text{WL}}{2}\text{x}-\frac{\text{W}}{2}\text{x}^{2}$
Find the point at which M is maximum in each case.

A and B are two events such that $\text{P}(\text{A})=\frac{1}{2},\text{P}(\text{B})=\frac{1}{3}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{4}.$ Find:

$\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$
$\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)$
$\text{P}\Big(\frac{\text{A}'}{\text{B}}\Big)$
$\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)$

A firm manufacturing two types of electric items, $A$ and $B,$ can make a profit of $Rs. 20$ per unit of $A$ and $Rs. 30$ per unit of $B.$ Each unit of $A$ requires $3$ motors and $4$ transformers and each unit of $B$ requires $2$ motors and $4$ transformers. The total supply of these per month is restricted to $210$ motors and $300$ transformers. Type $B$ is an export model requiring a voltage stabilizer which has a supply restricted to $65$ units per month. Formulate the linear programing problem for maximum profit and solve it graphically.

Let $A = R -\{3\}, B = R -\{1\}$. If $f: A \rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3} \forall x \in A$. Then, show that $f$ is bijective.

Solve the following differential equations:
$2(\text{y}+3)-\text{xy}\frac{\text{dy}}{\text{dx}}=0,\text{y}(1)=-2$

Evaluate the definite integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\sin2\text{x}\tan^{-1}(\sin\text{x})\text{dx}$