Question

If a and b are fixed non-zero constants, then the derivative of \(\frac{a}{x^4}-\frac{b}{x^2}\) + cos x is ma + nb - p where

• A. m = 4x³; \(n=\frac{-2}{x^3}\); p = sinx
• B. \(m =\frac{-4}{x^5}\); \(n=\frac{2}{x^3}\); p = sinx
• C. \(m =\frac{-4}{x^5}\); \(n=\frac{2}{x^3}\); p = -sinx
• D. m = 4x³; \(n=\frac{2}{x^3}\); p = -sinx

Answer 1

The Correct Option is B

We are given the function:
\[ \frac{a}{x^4} - \frac{b}{x^2} + \cos x \]
We need to find the derivative of this function.
Step 1: Differentiate \( \frac{a}{x^4} \)
Using the power rule, the derivative of \( \frac{a}{x^4} \) is:
\[ \frac{d}{dx} \left( \frac{a}{x^4} \right) = -\frac{4a}{x^5} \]
Thus, the first term gives: \( m = -\frac{4a}{x^5} \).
Step 2: Differentiate \( \frac{b}{x^2} \)
Using the power rule, the derivative of \( \frac{b}{x^2} \) is:
\[ \frac{d}{dx} \left( \frac{b}{x^2} \right) = -\frac{2b}{x^3} \]
Thus, the second term gives: \( n = -\frac{2b}{x^3} \).
Step 3: Differentiate \( \cos x \)
The derivative of \( \cos x \) is:
\[ \frac{d}{dx} (\cos x) = -\sin x \]
Thus, the third term gives: \( p = -\sin x \).
Step 4: Combine the results
Thus, the derivative of the given function is:
\[ m = -\frac{4a}{x^5}, \quad n = -\frac{2b}{x^3}, \quad p = -\sin x \]

Therefore, the correct answer is(B): \( m = -\frac{4a}{x^5} \), \( n = \frac{2b}{x^3} \), \( p = -\sin x \)

Answer 2

Let \(y = \frac{a}{x^4} - \frac{b}{x^2} + \cos x\). 

We can rewrite this as \(y = ax^{-4} - bx^{-2} + \cos x\).

Now, let's find the derivative of \(y\) with respect to \(x\):

\(\frac{dy}{dx} = a(-4x^{-5}) - b(-2x^{-3}) - \sin x\)

\(\frac{dy}{dx} = -\frac{4a}{x^5} + \frac{2b}{x^3} - \sin x\)

\(\frac{dy}{dx} = a(-\frac{4}{x^5}) + b(\frac{2}{x^3}) - \sin x\)

Given that \(\frac{dy}{dx} = ma + nb - p\), we can identify \(m\), \(n\), and \(p\).

\(m = -\frac{4}{x^5}\)

\(n = \frac{2}{x^3}\)

\(p = \sin x\)

Therefore, the correct option is:

\(m = -\frac{4}{x^5}\), \(n = \frac{2}{x^3}\), \(p = \sin x\)