Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate the following functions with respect to x:
$\frac{\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}}{\sqrt{\text{x}^2+1}-\sqrt{\text{x}^2-1}}$
✓

Answer

We have, $\frac{\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}}{\sqrt{\text{x}^2+1}-\sqrt{\text{x}^2-1}}$
By rationalising we get,
$\frac{\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}}{\sqrt{\text{x}^2+1}-\sqrt{\text{x}^2-1}}\times\frac{\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}}{\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}}$
$=\frac{\big(\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}\big)^2}{\big(\sqrt{\text{x}^2+1}\big)^2-\big(\sqrt{\text{x}^2-1}\big)^2}$
$=\frac{\big(\sqrt{\text{x}^2+1}\big)^2+\big(\sqrt{\text{x}^2-1}\big)^2+2\big(\sqrt{\text{x}^2+1}\big)\big(\sqrt{\text{x}^2-1}\big)}{\text{x}^2+1-\text{x}^2+1}$
$=\frac{\text{x}^2+1+\text{x}^2-1+2\sqrt{\text{x}^4-1}}{2}$
$=\frac{2\text{x}^2+2\sqrt{\text{x}^4-1}}{2}$
$=\text{x}^2+\sqrt{\text{x}^4-1}$
Now, Let $\text{y}=\text{x}^2+\sqrt{\text{x}^4-1}$
Differentiate it with respect to x we get,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{x}^2+\sqrt{\text{x}^4-1}\big)$
$=2\text{x}+\frac{1}{2\sqrt{\text{x}^4-1}}\times\frac{\text{d}}{\text{dx}}(\text{x}^4-1)$
$=2\text{x}+\frac{1}{2\sqrt{\text{x}^4-1}}\times(4\text{x}^3)$
$=2\text{x}+\frac{2\text{x}^3}{\sqrt{\text{x}^4-1}}$

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 "Solve the Following Question.(5 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Coloured balls are distributed in four boxes as shown in the following table:
Box
Colour
Black
White
Red
Blue
I
II
III
IV
3
2
1
4
4
2
2
3
5
2
3
1
6
2
1
5
A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III.
→
Solve the following differential equation
$\text{x}\frac{\text{dy}}{\text{dx}}+1=0;\text{y}(-1)=0$
→
Differentiate the following functions with respect to x:
$\log\Big(\frac{\text{x}^2+\text{x}+1}{\text{x}^3-\text{x}+1}\Big)$
→
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
 
 
Product A
Product B
Weekly capacity
Department 1
3
2
130
Department 2
4
6
260
Selling price per unit
Rs. 25
Rs. 30
 
Labour cost per unit
Rs. 16
Rs. 20
 
Raw material cost per unit
Rs. 4
Rs. 4
 
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
→
Solve the following systems of linear equations by cramer's rule:
x - 4y - z = 11,
2x - 5y + 2z = 39,
-3x + 2y + z = 1
→
Evaluate the following integrals as limit of sum:$\int\limits^{2}_{0}\big(\text{x}^2+2\big)\text{dx}$
→
Solve the following determinant equations:
$\begin{vmatrix}\text{x}+\text{a}&\text{x}&\text{x}\\\text{x}&\text{x}+\text{a}&\text{x}\\\text{x}&\text{x}&\text{x}+\text{a}\end{vmatrix}=0,\text{a}\neq0$
→
Let $\text{f}:[-1,\infty)\rightarrow[-1,\infty)$ be given by $f(x) = (x + 1)^2 - 1,  \text{x}\geq-1.$ Show that f is invertible. Also, find the set $S = \{x : f(x) = f^{-1}(x)\}.$
→
Solve the following systems of homogeneous linear equations by matrix method:
$x + y - z = 0$
$x - 2y + z = 0$
$3x + 6y - 5z = 0$
→
Show that $\text{f}(\text{x})=\sin\text{x}$ is increasing on $\Big(0,\frac{\pi}{2}\Big)$ and decreasing on $\Big(\frac{\pi}{2},\pi\Big)$ and neither increasing nor decreasing in $(0,\pi).$
→