
The value of \[\left( {\dfrac{d}{{dx}}} \right)\left( {\dfrac{1}{{{x^4}\sec x}}} \right)\] is equal to:
A. \[\dfrac{{[4\cos x - x\sin x]}}{{{x^5}}}\]
B. \[\dfrac{{ - {{[x\sin x + 4\cos x]}^5}}}{{{x^5}}}\]
C. \[\dfrac{{[4\cos x + x\sin x]}}{{{x^5}}}\]
D. None of these
Answer
162.9k+ views
Hint:
First we will simplify the given expression by using trigonometry ratios. Then we will apply the quotient rule of derivative to find the derivative of the given expression.
Formula Used:
\[\dfrac{d}{{dx}}\left[ {\dfrac{{f(x)}}{{g(x)}}} \right] = \dfrac{{\dfrac{d}{{dx}}f(x) \cdot g(x) - f(x)\dfrac{d}{{dx}}g(x)}}{{{{(g(x))}^2}}}\]
\[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\].
\[\cos {\rm{ x = }}\dfrac{1}{{\sec x}}\]
Complete step-by-step answer:
We have been given the function \[\left( {\dfrac{d}{{dx}}} \right)\left( {\dfrac{1}{{{x^4}\sec x}}} \right)\]
Let \[y = \dfrac{1}{{{x^4}\sec x}}\]
Use the property of \[\cos {\rm{ x}}\]:
\[y = \dfrac{{(\cos x)}}{{{x^4}}}\]
Use quotient rule:
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{{x^4}\dfrac{d}{{dx}}\cos x - \cos x\dfrac{{d{x^4}}}{{dx}}}}{{{{\left( {{x^4}} \right)}^2}}}\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\left( {{x^4}( - \sin x) - \cos x 4{x^3}} \right)}}{{{x^8}}}\]
We will take \[{x^3}\] common from numerator and divide it to \[{x^8}\] in denominator,
\[ = \dfrac{{ - (x\sin x + 4\cos x)}}{{{x^5}}}\]
Hence, option B is correct.
Note:
Students often make mistakes to find the derivative of \[\cos x\]. They forgot to put a negative sign of the derivative of \[\cos x\]. The derivative of \[\cos x\] is \[ - \sin x\].
First we will simplify the given expression by using trigonometry ratios. Then we will apply the quotient rule of derivative to find the derivative of the given expression.
Formula Used:
\[\dfrac{d}{{dx}}\left[ {\dfrac{{f(x)}}{{g(x)}}} \right] = \dfrac{{\dfrac{d}{{dx}}f(x) \cdot g(x) - f(x)\dfrac{d}{{dx}}g(x)}}{{{{(g(x))}^2}}}\]
\[\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}\].
\[\cos {\rm{ x = }}\dfrac{1}{{\sec x}}\]
Complete step-by-step answer:
We have been given the function \[\left( {\dfrac{d}{{dx}}} \right)\left( {\dfrac{1}{{{x^4}\sec x}}} \right)\]
Let \[y = \dfrac{1}{{{x^4}\sec x}}\]
Use the property of \[\cos {\rm{ x}}\]:
\[y = \dfrac{{(\cos x)}}{{{x^4}}}\]
Use quotient rule:
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{{x^4}\dfrac{d}{{dx}}\cos x - \cos x\dfrac{{d{x^4}}}{{dx}}}}{{{{\left( {{x^4}} \right)}^2}}}\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\left( {{x^4}( - \sin x) - \cos x 4{x^3}} \right)}}{{{x^8}}}\]
We will take \[{x^3}\] common from numerator and divide it to \[{x^8}\] in denominator,
\[ = \dfrac{{ - (x\sin x + 4\cos x)}}{{{x^5}}}\]
Hence, option B is correct.
Note:
Students often make mistakes to find the derivative of \[\cos x\]. They forgot to put a negative sign of the derivative of \[\cos x\]. The derivative of \[\cos x\] is \[ - \sin x\].
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NEET 2025 – Every New Update You Need to Know

Verb Forms Guide: V1, V2, V3, V4, V5 Explained

NEET Total Marks 2025

1 Billion in Rupees
