MHT CET · Maths · Differentiation
For \(x>1\), if \((2 x)^{2 y}=4 e^{2 x-2 y}\), then \(\left(1+\log _e 2 x\right)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to
- A \(x \log _{\mathrm{e}} 2 x\)
- B \(\log _e 2 x\)
- C \(\frac{x \log _{\mathrm{e}} 2 x+\log _{\mathrm{e}} 2}{x}\)
- D \(\frac{x \log _{\mathrm{e}} 2 x-\log _{\mathrm{e}} 2}{x}\)
Answer & Solution
Correct Answer
(D) \(\frac{x \log _{\mathrm{e}} 2 x-\log _{\mathrm{e}} 2}{x}\)
Step-by-step Solution
Detailed explanation
\((2 x)^{2 y}=4 \cdot e^{2 x-2 y}\)
\(\Rightarrow 2 y \log 2 x=\log 4+2 x-2 y\)
\(\Rightarrow y=\frac{x+\log 2}{1+\log 2 x}\)
\(\begin{aligned} & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{(1+\log 2 x)-(x+\log 2) \cdot \frac{1}{2 x} \cdot 2}{(1+\log 2 x)^2} \\ & \Rightarrow(1+\log 2 x)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x \log 2 x-\log 2}{x}\end{aligned}\)
\(\Rightarrow 2 y \log 2 x=\log 4+2 x-2 y\)
\(\Rightarrow y=\frac{x+\log 2}{1+\log 2 x}\)
\(\begin{aligned} & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{(1+\log 2 x)-(x+\log 2) \cdot \frac{1}{2 x} \cdot 2}{(1+\log 2 x)^2} \\ & \Rightarrow(1+\log 2 x)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x \log 2 x-\log 2}{x}\end{aligned}\)
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 then at = is ….MHT CET 2019 Easy
- If the angles \(\mathrm{A}, \mathrm{B}\) and C of a triangle ABC are in the ratio \(2: 3: 7\) respectively, then the sides a, b and c are respectively in the ratioMHT CET 2024 Medium
- The equation of the plane through the point \((2,-1,-3)\) and parallel to the lines \(\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}\) and \(\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}\) isMHT CET 2024 Medium
- \(\frac{1^{2}}{2}+\frac{1^{2}+2^{2}}{3}+\frac{1^{2}+2^{2}+3^{2}}{4}+\frac{1^{2}+2^{2}+3^{2}+4^{2}}{5}+\ldots \ldots \ldots \ldots\) upto 8 terms \(=\)MHT CET 2020 Medium
- \(\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x=\)MHT CET 2020 Easy
- The differential equation of \(y=\mathrm{e}^x\left(\mathrm{a}+\mathrm{b} x+x^2\right)\) isMHT CET 2024 Easy
More PYQs from MHT CET
- Heat of combustion of \(\mathrm{C}_{(\mathrm{s})}, \mathrm{H}_{2(\mathrm{~g})}\) and \(\mathrm{C}_{2} \mathrm{H}_{6(\mathrm{~g})}\) are \(-x_{1},-x_{2}\) and \(-x_{3}\) respectively. Hence heat of formation of \(\mathrm{C}_{2} \mathrm{H}_{6(\mathrm{~g})}\) isMHT CET 2020 Easy
- When the tension in string is increased by \(3 \mathrm{~kg} \omega \mathrm{t}\), the frequency of the fundamental mode increases in the ratio \(2: 3\). The initial tension in the string isMHT CET 2024 Medium
- Specific heats of an ideal gas at constant pressure and volume are denoted by \(C_p\) and \(C_v\) respectively. If \(\gamma=\frac{C_p}{C_v}\) and \(R\) is the universal gas constant then \(\mathrm{C}_{\mathrm{v}}\) is equal toMHT CET 2021 Easy
- Diosgenin obtained from yam plant (Dioscorea) is used in the manufacturing of_________.MHT CET 2022 Medium
- If
\(\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{\mathrm{a}}{2}(x-|x|), & \text { for } x \lt 0 \ 0, & \text { for } x=0 \ b x^2 \end{array}\right.\) \(\sin \left(\frac{1}{x}\right), \text { for } x\gt0\) is continuous at \(x=0\), thenMHT CET 2024 Medium - The order and degree of the differential equation \(\sqrt{\frac{d y}{d x}}-4 \frac{d y}{d x}-7 x=0\) are respectively.MHT CET 2021 Easy