Question

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 1

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 \[ - \sin x\].