MHT CET · Maths · Differentiation
Derivative of \(\sin ^2 x\) with respect to \(\mathrm{e}^{\cos x}\) is
- A \(2 \sin x \cos ^2 x \mathrm{e}^{\cos x}\)
- B \(\frac{2 \cos x}{\mathrm{e}^{\cos x}}\)
- C \(\frac{2 \sin x}{\mathrm{e}^{\cos x}}\)
- D \(\frac{-2 \cos x}{e^{\cos x}}\)
Answer & Solution
Correct Answer
(D) \(\frac{-2 \cos x}{e^{\cos x}}\)
Step-by-step Solution
Detailed explanation
Let \(\mathrm{u}=\sin ^2 x, \mathrm{v}=\mathrm{e}^{\cos x}\)
\(\mathrm{u}=\sin ^2 x\)
Differentiating w.r.t. \(x\), we get
\(\frac{\mathrm{du}}{\mathrm{~d} x}=2 \sin x \cdot \cos x\)
Consider, \(\mathrm{v}=\mathrm{e}^{\cos x}\)
Differentiating w.r.t. \(x\), we get
\(\begin{aligned}
& \therefore \quad \frac{\mathrm{dv}}{\mathrm{~d} x}=-\mathrm{e}^{\cos x} \cdot \sin x \\
& \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 \sin x \cdot \cos x}{-e^{\cos x} \cdot \sin x} \\
& =\frac{-2 \cos x}{\mathrm{e}^{\cos x}}
\end{aligned}\)
\(\mathrm{u}=\sin ^2 x\)
Differentiating w.r.t. \(x\), we get
\(\frac{\mathrm{du}}{\mathrm{~d} x}=2 \sin x \cdot \cos x\)
Consider, \(\mathrm{v}=\mathrm{e}^{\cos x}\)
Differentiating w.r.t. \(x\), we get
\(\begin{aligned}
& \therefore \quad \frac{\mathrm{dv}}{\mathrm{~d} x}=-\mathrm{e}^{\cos x} \cdot \sin x \\
& \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 \sin x \cdot \cos x}{-e^{\cos x} \cdot \sin x} \\
& =\frac{-2 \cos x}{\mathrm{e}^{\cos x}}
\end{aligned}\)
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 \(A^{-1}=\frac{-1}{2}\left[\begin{array}{cc}5 & 8 \\ -1 & 2\end{array}\right]\), then \(2 A+I_2=\), where \(I_2\) is a unit matrix of order 2MHT CET 2021 Medium
- There are 2 shelves. One shelf has 5 Physics and 3 Bilogy books and other has 4 Physics and 2 Biology books. Then probability of drawing a Physics book isMHT CET 2022 Medium
- If the angle between the planes and is thenMHT CET 2017 Easy
- If \(\mathrm{y}=a^x \cdot \mathrm{~b}^{2 x-1}\), then \(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{d} x^2}\) is equal toMHT CET 2025 Medium
- If
\(f(x)= \begin{cases}a x^2+b x+1 & \text { if }|2 x-3| \geq 2 \ 3 x+2 & \text {; if } \end{cases}\) \(\frac{1}{2} < x < \frac{5}{2}\) is continuous on its domain, then \(a+b\) has the valueMHT CET 2022 Medium - \(\int \frac{\mathrm{d} x}{2 \mathrm{e}^{2 x}+3 \mathrm{e}^x+1}=\)MHT CET 2025 Medium
More PYQs from MHT CET
- What will be the length of a dsDNA strand, if it contains 100 base pairs?MHT CET 2023 Hard
- If \(\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x\) and \(\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))\), then \(\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}\) isMHT CET 2023 Hard
- A disc of radius \(R\) and thickness \(\frac{R}{6}\) has moment of inertia I about an axis passing through its centre and perpendicular to its plane. Disc is melted and recast into a solid sphere. The moment of inertia of a sphere about its diameter isMHT CET 2023 Medium
- Pericardial fluid is present between _________ of heart.MHT CET 2022 Medium
- DNA was first isolated from the nuclei of pus cells by I and named as II .MHT CET 2023 Easy
- In Young's double slit experiment, the distance between screen and aperture is 1 m . The slit width is 2 mm . Light of \(6000 Å\) is used. If a thin glass plate ( \(\mu=1.5\) ) of thickness 0.04 mm is placed over one of the slits, then there will be a lateral displacement of the fringes byMHT CET 2025 Medium