Download App

Differentiation — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferentiation5 Marks
Question
If $e^x + x^y = e^{x+y},$ prove that $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$
✓

Answer

Here,
$e^x + e^y = e^{x+y} ......(i)$
Differentiating both the sides using chain rule,
$\frac{\text{d}}{\text{dx}}(\text{e}^\text{x})+\frac{\text{d}}{\text{dx}}(\text{e}^\text{y})=\frac{\text{d}}{\text{dx}}(\text{e}^{\text{x}+\text{y}})$
$\text{e}^{\text{x}}+\text{e}^{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}\frac{\text{d}}{\text{dx}}(\text{x}+\text{y})$
$\text{e}^{\text{x}}+\text{e}^{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}\Big[1+\frac{\text{d}}{\text{dx}}\Big]$
$\text{e}^{\text{y}}\frac{\text{dy}}{\text{dx}}-\text{e}^{\text{x}+\text{y}}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}-\text{e}^\text{x}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{e}^{\text{x}+\text{y}}-\text{e}^\text{x}}{\text{e}^\text{y}-\text{e}^{\text{x}+\text{y}}}$
$=\Big(\frac{\text{e}^\text{x}+\text{e}^\text{x}-\text{e}^\text{x}}{\text{e}^\text{y}-\text{e}^\text{x}-\text{e}^\text{y}}\Big)$
$[$Using equation $(i)]$
$\frac{\text{dy}}{\text{dx}}=-\text{e}^{\text{y}-\text{x}}$
$\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following:
$\begin{bmatrix}1&2&3\end{bmatrix}\begin{bmatrix}1&0&2\\2&0&1\\0&1&2\end{bmatrix}\begin{bmatrix}2\\4\\6\end{bmatrix}$
→
Differentiate the following functions with respect to x:
$\text{x}^{(\sin\text{x}-\cos\text{x})}+\frac{\text{x}^2-1}{\text{x}^2+1}$
→
On a multiple choice examination with three possible answers (out of which only one is correct) fo each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
→
Write a value of $\int\text{e}^{\text{ax}}\cos\text{bx}\text{ dx}$
→
Find the distance between the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})+7=0$ and $\vec{\text{r}}\cdot(2\hat{\text{i}}+4\hat{\text{j}}+6\hat{\text{k}})+7=0$
→
Find the shortest distance between the given lines. $\vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$, $\vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(-2 \hat{i}+3 \hat{j}+8 \hat{k})$
→
If A and B are two events such that $2\text{P(A)}=\text{P(B)}=\frac{5}{13}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{2}{5}$ find $\text{P}(\text{A}\cap\text{B}).$
→
Find the equatoion of the passing through the points (1, -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = 9.
→
Evaluate the following intregals:
$\int\frac{5\text{x}}{(\text{x}+1)(\text{x}^2-4)}\text{dx}$
→
Show that $\text{y}=\text{A}\cos2\text{x}+\text{B}\sin2\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$
→