Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\frac{\text{e}^{\text{ax}}\sec\text{x}\times\log\text{x}}{\sqrt{1-2\text{x}}}$
✓

Answer

Here,
$\text{y}=\frac{\text{e}^{\text{ax}}\sec\text{x}\times\log\text{x}}{\sqrt{1-2\text{x}}}\ .....(\text{i})$
$\Rightarrow\text{y}=\frac{\text{e}^{\text{ax}}\times\sec^\text{x}\times\log\text{x}}{(1-2\text{x})^\frac{1}{2}}$
Taking log on both the sides,
$\log\text{y}=\log\text{e}^{\text{ax}}+\log{\sec\text{x}}+\log\log\text{x}-\frac{1}{2}\log(1-2\text{x}) \\ \begin{bmatrix} \text{Since}, \log\Big(\frac{\text{A}}{\text{B}}\Big)=\log\text{A}-\log\text{B},\\ \log(\text{AB})=\log\text{A}+\log\text{B} \end{bmatrix}$
$\log\text{y}=\text{ax}+\log{\sec\text{x}}+\log\log\text{x}-\frac{1}{2}\log(1-2\text{x}) \\ \big[\text{Since}, \log\text{a}^\text{b}=\text{b}\log\text{a and }\log_\text{e}\text{e}=1\big]$
Differentiating it with respect to x using chain rule,
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\text{ax})+\frac{\text{d}}{\text{dx}}(\log\sec\text{x})+\frac{\text{d}}{\text{dx}}(\log\log\text{x})-\frac{1}{2}\log(1-2\text{x})$
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{a}+\frac{1}{\sec\text{x}}\frac{\text{d}}{\text{dx}}(\sec\text{x})+\frac{1}{\log\text{x}}\frac{\text{d}}{\text{dx}}-\frac{1}{2}\Big(\frac{1}{1-2\text{x}}\Big)\frac{\text{d}}{\text{dx}}(1-2\text{x})$
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{a}+\frac{\sec\text{x}\tan\text{x}}{\sec\text{x}}+\frac{1}{(\log\text{x})}\Big(\frac{1}{\text{x}}\Big)-\frac{1}{2}\Big(\frac{1}{1-2\text{x}}\Big)(-2)$
$\frac{\text{dy}}{\text{dx}}=\text{y}\Big[\text{a}+\tan\text{x}+\frac{1}{\text{x}\log\text{x}}+\frac{1}{1-2\text{x}}\Big]$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{e}^{\text{ax}}\sec\text{x}\log\text{x}}{\sqrt{1-2\text{x}}}\Big[\text{a}+\tan\text{x}+\frac{1}{\text{x}\log\text{x}}+\frac{1}{1-2\text{x}}\Big]$
[Using equation (i)]

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

How should we choose two numbers, each greater than or equal to −2, whose sum so that the sum of the first and the cube of the second is minimum?
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\tan\text{x}}}{\sqrt{\tan\text{x}}+\sqrt{\cot\text{x}}}\text{ dx}$
→
Find the area of the region $\{(\text{x},\text{y}):\text{x}^2+\text{y}^2\leq4,\text{x}+\text{y}\geq2\}$
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\text{e}^{2\text{x}}\sin\Big(\frac{\pi}{4}+{\text{x}}\Big)\text{dx}$
→
If$ y = P e^{ax}+ Q e^{bx},$ show that
$\frac{\text{d}^2\text{y}}{\text{dx}^2}-(\text{a}+\text{b})\frac{\text{dy}}{\text{dx}}+\text{aby}=0$
→
Bag $A$ contains $3$ red and $5$ black balls, while bag $B$ contains $4$ red and $4$ black balls. Two balls are transferred at random from bag $A$ to bag $B$ and then a ball is drawn from bag $B$ at random. If the ball drawn from bag $B$ is found to be red, find the probability that two red balls were transferred from $A$ to $B.$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}-1}{(\text{x}+1)^2(\text{x}+2)}\ \text{dx}$
→
The scalar product of the vector $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ with the unit vector along the sum of vectors $2\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}$ and $\lambda\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$ is equal to one. Find the value of λ.
→
Given $\text{A}=\begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix},\text{B}=\begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$ , find $BA$ and use this to solve the system of equations $y + 2z = 7, x - y = 3, 2x + 3y + 4z = 17$
→
If $e^x + x^y = e^{x+y},$ prove that $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$
→