Find dy dx , of the function y^x = x^y - Vidyadip

Question

Find $\frac{{dy}}{{dx}}$, of the function $y^x = x^y$

✓

Answer

Given:$y^x = x^y$
$\Rightarrow {x^y} = {y^x}$
$\Rightarrow \log {x^y} = \log {y^x}$
$ \Rightarrow y\log x = x\log y$
$\Rightarrow \frac{d}{{dx}}\left( {y\log x} \right) = \frac{d}{{dx}}\left( {x\log y} \right)$
$\Rightarrow y.\frac{1}{x} + \log x.\frac{{dy}}{{dx}} = x.\frac{1}{y}\frac{{dy}}{{dx}} + \log y.1$
$\Rightarrow \left( {\log x - \frac{x}{y}} \right)\frac{{dy}}{{dx}} = \log y - \frac{y}{x}$
$\Rightarrow \left( {\frac{{y\log x - x}}{y}} \right)\frac{{dy}}{{dx}} = \frac{{x\log y - y}}{x}$
$\Rightarrow \frac{{dy}}{{dx}} = \frac{{y\left( {x\log y - y} \right)}}{{x\left( {y\log x - x} \right)}}$

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 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

Given two independent events A and B such that P(A) = 0.3 and P(B) = 0.6. Find
$\text{P}(\text{A}\cup\text{B})$

Determine whether the relation is reflexive, symmetric and transitive:
Relation R in the set A of human beings in a town at a particular time given by
R = {(x, y) : x is wife of y}

A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that
all are white?

Find the interval where function $f(x)=x^3-3 x$ is decreasing.

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among 100 students, what is the probability that:
You both enter the different sections?

Show that $\text{f}(\text{x})=\sin\text{x}$ is an increasing function on $\Big(\frac{-\pi}{2},\frac{\pi}{2}\Big).$

Find $\vec{\text{a}}.\vec{\text{b}}$ when
$\vec{\text{a}}=\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}-2\hat{\text{k}}$

Evaluate the following integrals:
$\int\limits^1_{-2}\frac{|\text{x}|}{\text{x}}\text{ dx}$

A coin is tossed three times. Find the mean for number of heads.

Find the components along the coordinate axis of the position vector of the following point:S(4,-3)