Download App

Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability2 Marks
Question
Find $\frac{dy}{dx}$ of the function xy + yx = 1
✓

Answer

Given: xy + yx = 1
Let y = xy + yx = 1
Let u = xy and v = yx
Then, u + v = 1
$\Rightarrow \frac{d u}{d x}+\frac{d v}{d x}=0$ 
For, u = xy 
Taking log on both sides, we get
log u = log xy 
$\Rightarrow \log u=y \cdot \log (x)$ 
Now, differentiating both sides with respect to x
$\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{u})=\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{y} \cdot \log (\mathrm{x})]$ 
$\Rightarrow \frac{1}{\mathrm{u}} \frac{\mathrm{du}}{\mathrm{dx}}=\left\{\mathrm{y} \cdot \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})+\log \mathrm{x} \cdot \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{y})\right\}$ 
$\Rightarrow \frac{d u}{d x}=u\left[y \cdot \frac{1}{x}+\log x \cdot\left(\frac{d y}{d x}\right)\right]$ 
$\Rightarrow \frac{\mathrm{du}}{\mathrm{dx}}=\mathrm{x}^{\mathrm{y}}\left[\frac{\mathrm{y}}{\mathrm{x}}+\log \mathrm{x} \cdot\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]$ 
For v = yx 
Taking log on both sides, we get
log v = log yx  
$\Rightarrow \log v=x \cdot \log (y)$ 
Now, differentiate both sides with respect to x
$\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{v})=\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{x} \cdot \log (\mathrm{y})]$ 
$\Rightarrow \frac{1}{\mathrm{v}} \frac{\mathrm{dv}}{\mathrm{dx}}=\left\{\mathrm{x} \cdot \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{y})+\log \mathrm{y} \cdot \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}\right\}$ 
$\Rightarrow \frac{d v}{d x}=v\left[x \cdot \frac{1}{y} \cdot \frac{d y}{d x}+\log y \cdot\left(\frac{d x}{d x}\right)\right]$ 
$\Rightarrow \frac{d v}{d x}=y^{x}\left[\frac{x}{y} \cdot \frac{d y}{d x}+\log y\right]$ 
because, $\frac{d u}{d x}+\frac{d v}{d x}=0$ 
So, $\mathrm{x}^{\mathrm{y}}\left[\frac{\mathrm{y}}{\mathrm{x}}+\log \mathrm{x} \cdot\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]+\mathrm{y}^{\mathrm{x}}\left[\frac{\mathrm{x}}{\mathrm{y}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\log \mathrm{y}\right]=0$ 
$\Rightarrow\left(\mathrm{x}^{\mathrm{y}} \log \mathrm{x}+\mathrm{xy}^{\mathrm{x}-1}\right) \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\left(\mathrm{yx}^{\mathrm{y}-1}+\mathrm{y}^{\mathrm{x}} \log \mathrm{y}\right)=0$ 
$\Rightarrow\left(\mathrm{x}^{\mathrm{y}} \log \mathrm{x}+\mathrm{xy}^{\mathrm{x}-1}\right) \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=-\left(\mathrm{yx}^{\mathrm{y}-1}+\mathrm{y}^{\mathrm{x}} \log \mathrm{y}\right)$ 
$\frac{d y}{d x}=-\frac{\left(y x^{y-1}+y^{x} \log y\right)}{\left(x^{y} \log x+x y^{x-1}\right)}$

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 "2 Marks" 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

Write the principal value of $\sin^{-1}\Big\{\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big\}$
→
Prove that $(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2}$, if and only if $\vec{a}, \vec{b}$ are perpendicular, given $\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}$.
→
If $\text{A}=\begin{bmatrix}\cos\text{x}&-\sin\text{x}\\\sin\text{x}&\cos\text{x}\end{bmatrix},$ find AAT.
→
In the matrix A = $\left[\begin{array}{cccc} {2} & {5} & {19} & {-7} \\ {35} & {-2} & {\frac{5}{2}} & {12} \\ {\sqrt{3}} & {1} & {-5} & {17} \end{array}\right]$ write:
  1. The order of the matrix,
  2. The number of elements,
  3. Write the elements a13, a21, a33, a24, a23.
→
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
→
If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=5$ and $\vec{\text{a}}.\vec{\text{b}}=2,$ find $\big|\hat{\text{a}}-\hat{\text{b}}\big|.$
→
If $\text{A}=\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix},$ find A2.
→
Find the second-order derivatives of the function x20
→
Write a unit vector in the direction of the sum of the vectors $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+\text{y}\hat{\text{j}}-7\hat{\text{k}}$.
→
For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ write the value of $\big(\vec{\text{a}}.\vec{\text{b}}\big)^2+\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2$ in terms of their magnitudes.
→