Download App

Differentiation — Maths STD 12 Science — Question

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

Answer

Let $\text{y}=\sqrt{\frac{1+\sin\text{x}}{1-\sin\text{x}}}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\frac{1+\sin\text{x}}{1-\sin\text{x}}\Big)^{\frac{1}{2}}$
$=\frac{\text{1}}{\text{2}}\Big(\frac{1+\sin\text{x}}{1-\sin\text{x}}\Big)^{\frac{1}{2}-1}\frac{\text{d}}{\text{dx}}\Big(\frac{1+\sin\text{x}}{1-\sin\text{x}}\Big)$
$=\frac{\text{1}}{\text{2}}\Big(\frac{1+\sin\text{x}}{1-\sin\text{x}}\Big)^{\frac{1}{2}}\Big[\frac{(1-\sin\text{x})(\cos\text{x})-(1+\sin\text{x})(-\cos\text{x})}{(1-\sin\text{x})^2}\Big]$
$=\frac{\text{1}}{\text{2}}\frac{(1+\sin\text{x})^\frac{1}{2}}{(1-\sin\text{x})^\frac{1}{2}}\Big[\frac{\cos\text{x}-\cos\text{x}+\cos\text{x}\sin\text{x}+\sin\text{x}\cos\text{x}}{(1-\sin\text{x})^2}\Big]$
$=\frac{1}{2}\times\frac{2\cos\text{x}}{\sqrt{1+\sin\text{x}}(1-\sin\text{x})^\frac{3}{2}}$
$=\frac{\cos\text{x}}{\sqrt{1+\sin\text{x}}(1-\sin\text{x})^\frac{3}{2}}$
$=\frac{\cos\text{x}}{\sqrt{1+\sin\text{x}}\sqrt{1-\sin\text{x}}(1-\sin\text{x})}$
$=\frac{\cos\text{x}}{\sqrt{1-\sin^2\text{x}}(1-\sin\text{x})}$
$=\frac{\cos\text{x}}{\cos\text{x}(1-\sin\text{x})}$ $\big[\text{Using }1-\sin^2\text{x}=\cos^2\text{x}\big] $
$=\frac{1}{(1-\sin\text{x})}\times\frac{(1+\sin\text{x})}{(1+\sin\text{x})}$
Thus, $ \frac{\text{dy}}{\text{dx}}=\frac{1}{\cos^2\text{x}}+\frac{\sin\text{x}}{\cos^2\text{x}}$
$\frac{\text{dy}}{\text{dx}}=\sec^2\text{x}+\tan\text{x}\sec\text{x}$
$\frac{\text{dy}}{\text{dx}}=\sec\text{x}[\tan\text{x}+\sec\text{x}] $

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

Find the area of the region between the parabola $x = 4y - y^2$ and the line $x = 2y - 3$
→
Find the area of the region included between $y^2 = 9x$ and $y = x$.
→
Evaluate the following:
$\begin{pmatrix}\begin{bmatrix}1&3\\-1&-4 \end{bmatrix}+\begin{bmatrix}3&-2\\-1&1 \end{bmatrix}\end{pmatrix}\begin{bmatrix}1&3&5\\2&4&6 \end{bmatrix}$
→
A trust invested some money in two type of bonds. The first bond pays 10% interest and bond pays 12% interest. The trust received 2,800 as interest. However, if trust had interchanged money in bonds, they would have got 100 less as interest. Using matrix method, find the amount invested by the trust. Which value is reflected in this question?
→
If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.
→
Solve the differential equation $\frac{d y}{d x}+y \tan x=y^2$ $\sec x$.
→
Find the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line $\frac{\text{x}-1}{1}=\frac{2\text{y}+1}{2}=\frac{\text{z}+1}{-1}.$
→
Sketch the region bounded by the curves $y = x^2 + 2, y = x, x = 0$ and $x = 1$. Also find the area of this region.
→
The contents of three urns are as follows:
Urn 1 : 7 white, 3 black balls,
Urn 2 : 4 white, 6 black balls,
Urn 3 : 2 white, 8 black balls.
One of these urns is chosen at random with probabilities 0.20, 0.60 and 0.20 respectively. From the chosen urn two balls are drawn at random without replacement. If both these balls are white, what is the probability that these came from urn 3?
→
Find the particular solution of the differential equation$\text{e}^{x}\sqrt{1 - \text{y}^{2}} \text{ dx} + \frac{\text{y}}{\text{x}}\text{dy } = 0 $ given that y= 1 when x=0.
→