Question

Let \(f(x)=6\sqrt{x^5}^3\). If \(f'(x)=ax^p\), where a and p are constants, then the value of p is equal to

• A. \(\frac{3}{5}\)
• B. \(\frac{-2}{5}\)
• C. \(\frac{13}{2}\)
• D. \(\frac{-2}{3}\)
• E. \(\frac{2}{5}\)

Answer 1

The Correct Option is C

We are given that \( f(x) = 6\sqrt{x^5}^3 \), and we need to find the value of \( p \) such that \( f'(x) = ax^p \), where \( a \) and \( p \) are constants.

First, simplify the expression for \( f(x) \): 

\(f(x) = 6 \cdot \left( x^5 \right)^{3/2} = 6x^{15/2}\)

Now, differentiate \( f(x) \) with respect to \( x \):

\(f'(x) = \frac{d}{dx} \left( 6x^{15/2} \right) = 6 \cdot \frac{15}{2} x^{15/2 - 1} = 45x^{13/2}\)

Thus, we have \( f'(x) = 45x^{13/2} \), which is of the form \( ax^p \), where \( a = 45 \) and \( p = \frac{13}{2} \).

Therefore, the value of \( p \) is:

\(\frac{13}{2}\)