MHT CET · Maths · Differentiation
If \(\mathrm{f}(x)=\log _{x^2}(\log x)\), then at \(x=\mathrm{e}, \mathrm{f}^{\prime}(x)\) has the value
- A \(\frac{1}{\mathrm{e}^2}\)
- B \(\frac{1}{\mathrm{e}}\)
- C \(\mathrm{e}^{2 \cdot}\)
- D \(\frac{1}{2 \mathrm{e}}\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{2 \mathrm{e}}\)
Step-by-step Solution
Detailed explanation
Let \(y=\log _{x^2}(\log x)\)
\(\therefore \quad x^{2 y}=\log x\)
Differentiating w.r.t. \(x\), we get
\(\begin{aligned}
& x^{2 y}(\log x) \times 2 \times \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{x} \\
& \therefore \quad(\log x)^2 \times \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2 x} \\
& \quad \text { At } x=\mathrm{e}, \mathrm{f}^{\prime}(x)=\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2 \mathrm{e}}
\end{aligned} \quad \ldots\left[\because x^{2 y}=\log x\right]\)
\(\therefore \quad x^{2 y}=\log x\)
Differentiating w.r.t. \(x\), we get
\(\begin{aligned}
& x^{2 y}(\log x) \times 2 \times \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{x} \\
& \therefore \quad(\log x)^2 \times \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2 x} \\
& \quad \text { At } x=\mathrm{e}, \mathrm{f}^{\prime}(x)=\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2 \mathrm{e}}
\end{aligned} \quad \ldots\left[\because x^{2 y}=\log x\right]\)
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
- \(\int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{\mathrm{e}^{\sin ^2 x+\cos ^3 x}} \mathrm{~d} x=\)MHT CET 2023 Hard
- \(\int \frac{\mathrm{e}^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] \mathrm{d} x, x>0=\)MHT CET 2023 Hard
- If the acute angle between the lines \(x^{2}-4 x y+y^{2}=0\) is \(\tan ^{-1}(k)\), then \(\mathrm{k}=\)MHT CET 2020 Easy
- A lot of 100 bulbs contains 10 defective bulbs. Five bulbs are selected at random from the lot and are sent to retail store. Then the probability that the store will receive at most one defective bulb isMHT CET 2023 Easy
- The population of a town increases at a rate proportional to the population at that time. If the population increases from forty thousand to eighty thousand in 20 years, then the population in another 40 years will beMHT CET 2025 Medium
- The equation of the tangent to the parabola \(y^2=8 x\), which is parallel to the line \(4 x-y+3=0\) isMHT CET 2024 Medium
More PYQs from MHT CET
- A tuning fork gives 3 beats with \(50 \mathrm{~cm}\) length of sonometer wire. If the length of the wire is shortened by \(1 \mathrm{~cm}\), the number of beats is still the same. The frequency of the fork isMHT CET 2023 Medium
- In Haber's process of production of ammonia, \(\mathrm{K}_2 \mathrm{O}\) is used asMHT CET 2024 Easy
- Identify the molecule containing triple bond.MHT CET 2022 Medium
- An inductive coil has a resistance of \(100 \Omega\). When an a.c. signal of frequency \(1000 \mathrm{~Hz}\) is applied to the coil the voltage leads the current by \(45^{\circ}\). The inductance of the coil isMHT CET 2021 Easy
- If where , then A (adj A) = ….MHT CET 2019 Easy
- Air capacitor has capacitance of \(1 \mu \mathrm{~F}\). Now the space between two plates of capacitor is filled with two dielectrics as shown in figure. The capacitance of the capacitor is ( \(\mathrm{d}=\) distance between two plates, \(\mathrm{K}_1\) and \(\mathrm{K}_2\) are dielectric constants of two dielectrics respectively)
MHT CET 2025 Easy