KCET · Maths · Differentiation
If \(x^{x}=y^{y}\), then \(\frac{d y}{d x}\) is
- A \(-\frac{y}{x}\)
- B \(-\frac{x}{y}\)
- C \(1+\log \left(\frac{x}{y}\right)\)
- D \(\frac{1+\log x}{1+\log y}\)
Answer & Solution
Correct Answer
(D) \(\frac{1+\log x}{1+\log y}\)
Step-by-step Solution
Detailed explanation
Given, \(x^{x}=y^{y}\)
Taking log on both sides, we get
\[
x \log x=y \log y
\]
Differentiating w.r.t. \(y\), we get
\[
\begin{aligned}
&y \cdot \frac{1}{y} \cdot \frac{d y}{d x}+\log y \frac{d y}{d x}=x \frac{1}{x}+\log x \\
&\Rightarrow \quad \frac{d y}{d x}(1+\log y)=1+\log x \\
&\Rightarrow \quad \frac{d y}{d x}=\frac{1+\log x}{1+\log y}
\end{aligned}
\]
Taking log on both sides, we get
\[
x \log x=y \log y
\]
Differentiating w.r.t. \(y\), we get
\[
\begin{aligned}
&y \cdot \frac{1}{y} \cdot \frac{d y}{d x}+\log y \frac{d y}{d x}=x \frac{1}{x}+\log x \\
&\Rightarrow \quad \frac{d y}{d x}(1+\log y)=1+\log x \\
&\Rightarrow \quad \frac{d y}{d x}=\frac{1+\log x}{1+\log y}
\end{aligned}
\]
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(y=\left(\cos x^{2}\right)^{2}\), then \(\frac{d y}{d x}\) is equal toKCET 2021 Easy
- \(\int_{\pi / 6}^{\pi / 3} \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x\) is equal toKCET 2012 Easy
- If \( \cos \alpha, \cos \beta, \cos \gamma \) are the direction cosines of a vector \( \vec{a} \), then \( \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma \) is
equal toKCET 2016 Medium - If \(A = \{a, b, c, d, e, f\}\), then the number of subsets of A which contains at least \(2\) elements isKCET 2026 Easy
- If the conjugate of \((x+i y)(1-2 i)\) is \(1+i\), thenKCET 2012 Hard
- \(\int_{0}^{1} x(1-x)^{3 / 2} d x\) isKCET 2010 Hard
More PYQs from KCET
- If \(f(x) = \begin{cases} ax + 7 & \text{if } x < 1 \\ 3x - 1 & \text{if } x = 1 \\ \dfrac{b}{x + 3} & \text{if } x > 1 \end{cases}\) is continuous at \(x = 1\), thenKCET 2026 Easy
- Benzylamine is a stronger base than aniline becauseKCET 2012 Easy
- \( \int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} \mathrm{dx} \) is equal toKCET 2015 Medium
- Gynaecomastia is the symptom ofKCET 2013 Hard
- A solid sphere of mass \(m\) rolls down an inclined plane without slipping, starting from rest at the top of an inclined plane. The linear speed of the sphere at the bottom of the inclined plane is \(v\). The kinetic energy of the sphere at the bottom isKCET 2011 Hard
- \(\int \frac{\mathrm{dx}}{\mathrm{x}^2\left(\mathrm{x}^4+1\right)^{3 / 4}}\) equalsKCET 2025 Medium