MHT CET · Maths · Differentiation
If \(y\) is a function of \(x\) and \(\log (x+y)=2 x y\), then the value of \(y^{\prime}(0)\) is
- A 1
- B -1
- C 2
- D 0
Answer & Solution
Correct Answer
(A) 1
Step-by-step Solution
Detailed explanation
\(\log (x+y)=2 x y...(i)\)
Differentiating both sides w.r.t. \(x\), we get
\(\begin{aligned}
& \left(\frac{1}{x+y}\right)\left(1+\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=2\left(x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y\right) \\
& \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1-2 x y-2 y^2}{2 x^2+2 x y-1}
\end{aligned}\)
Putting \(x=0\) in (i), we get
\(\begin{aligned}
& y=1 \\
\therefore \quad & y^{\prime}(0)=\frac{1-0-2}{0+0-1}=1
\end{aligned}\)
Differentiating both sides w.r.t. \(x\), we get
\(\begin{aligned}
& \left(\frac{1}{x+y}\right)\left(1+\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=2\left(x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y\right) \\
& \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1-2 x y-2 y^2}{2 x^2+2 x y-1}
\end{aligned}\)
Putting \(x=0\) in (i), we get
\(\begin{aligned}
& y=1 \\
\therefore \quad & y^{\prime}(0)=\frac{1-0-2}{0+0-1}=1
\end{aligned}\)
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 \(I=\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x\), then value of \(I\) isMHT CET 2024 Medium
- If \(\mathrm{p}\) and \(\mathrm{q}\) are true statements and \(\mathrm{r}\) and \(s\) are false statements, then the truth values of the statement patterns \((p \wedge q) \vee r\) and \((\mathrm{p} \vee \mathrm{s}) \leftrightarrow(\mathrm{q} \wedge \mathrm{r})\) are respectivelyMHT CET 2023 Medium
- If \(\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 4 \\ 1 & 3 & 4\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}12 \\ 15 \\ 13\end{array}\right]\), then the value of \(x^2+y^2+z^2=\)MHT CET 2025 Medium
- The intercept of the latusrectum to the parabola \(y^{2}=4 a x\) are \(b\) and \(k\), then \(k\) is equal toMHT CET 2012 Easy
- The acute angle between the diagonals of a parallelogram whose vertices are \(\mathrm{A}(2,-1), \mathrm{B}(0,2), \mathrm{C}(2,3)\) and \(\mathrm{D}(4,0)\) isMHT CET 2025 Medium
- Which of the following statement has the truth value ' \(F^{\prime}\) ?MHT CET 2007 Easy
More PYQs from MHT CET
- Equal volumes of two gases, having their densities in the ratio of \(1: 16\) exert equal pressures on the walls of two containers. The ratio of their rms speeds \(\left(c_1: c_2\right)\) isMHT CET 2021 Easy
- The equation that represents the van't Hoff general solution equation isMHT CET 2016 Easy
- The acid which contains both -OH and -COOH groups isMHT CET 2017 Easy
- Which from following combinations represents water gas?MHT CET 2024 Hard
- In successive emission of \(\alpha\) -and \(\beta\) -particles, the number of \(\alpha\) -and \(\beta\) -particles should be emitted for the conversion of \({ }_{92} \mathrm{U}^{238}\) to \({ }_{82} \mathrm{~Pb}^{206}\) areMHT CET 2008 Medium
- What is the co-ordination number of a particle in fcc crystal lattice?MHT CET 2022 Easy