Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{e}^{\text{x}}+\text{e}^{\text{y}}=\text{e}^{\text{x}+\text{y}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{e}^{\text{x}}(\text{e}^\text{y}-1)}{\text{e}^{\text{y}}(\text{e}^{\text{x}}-1)}$ or $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$
✓

Answer

$\text{e}^\text{x}+\text{e}^\text{y}=\text{e}^{\text{x}+\text{y}}$
$\Rightarrow\text{e}^\text{x}+\text{e}^\text{y}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}\Big(1+\frac{\text{dy}}{\text{dx}}\Big)$
$\Rightarrow\text{e}^\text{x}+\text{e}^{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}+\text{e}^{\text{x}+\text{y}}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\text{e}^\text{y}\frac{\text{dy}}{\text{dx}}-\text{e}^{\text{x}+\text{y}}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}-\text{e}^{\text{x}}$
$\Rightarrow\frac{\text{dx}}{\text{dy}}(\text{e}^\text{y}-\text{e}^{\text{x}+\text{y}})=\text{e}^{\text{x}+\text{y}}-\text{e}^{\text{x}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{e}^{\text{x}+\text{y}}-\text{e}^{\text{x}}}{\text{x}^\text{y}-\text{e}^{\text{x}+\text{y}}}$
$=\frac{\text{e}^\text{x}(\text{e}^\text{y}-1)}{\text{e}^\text{y}({1-\text{e}}^\text{x})}$
$=-\frac{\text{e}^\text{x}(\text{e}^\text{y}-1)}{\text{e}^\text{y}(\text{e}^\text{x}-1)}$

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

Similar questions

For the following matrices verify the associativity of multiplication i.e., (AB) C = A(BC):
$\text{A}=\begin{bmatrix}1&2&0\\-1&0&1\end{bmatrix},\text{B}=\begin{bmatrix}1&0\\-1&2\\0&3\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1\\-1\end{bmatrix}$
→
If $\text{A}=\begin{bmatrix}3&-4\\1&1\\2&0\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&1&2\\1&2&4\end{bmatrix},$ then verify $(\text{BA})^2\neq\text{B}^2\text{A}^2.$
→
Find the equations of the two lines through the origin which intersect the line $\frac{\text{x}-3}{2}-\frac{\text{y}-3}{1}=\frac{\text{z}}{1}$ at angles of $\frac{\pi}{3}$ each.
→
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10 m.$ Find the dimensions of the window to admit maximum light through the whole opening
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(2\text{x}+1)(\text{x}^2-1)}\text{ dx}$
→
Differentiate the following functions with respect to x:
$\sin^2\{\log(2\text{x}+3)\}$
→
Of all the closed right circular cylindrical cans of volume $128\ \pi\ cm^3$ , find the dimensions of the can which has minimum surface area.
→
Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^3\text{x}\text{ dx}$
→
Find the intervals in which $\text{f}(\text{x})=\log(1+\text{x})-\frac{\text{x}}{1+\text{x}}$ is increasing or decreasing.
→
Solve the following systems of linear equations by cramer's rule:
3x + ay = 4,
2x + ay = 2, $\text{a}\neq0$
→