Question

for the function ( f(x)=2sqrt{x^5} ), find ( f(x) ) by rules. answer: ( f(x)= ) box × .

Explanation:

Step1: Rewrite the function

First, rewrite the square - root and the power of \(x\) in the function \(f(x) = 2\sqrt{x^{5}}\). Recall that \(\sqrt{a}=a^{\frac{1}{2}}\) and \((a^{m})^{n}=a^{mn}\). So, \(\sqrt{x^{5}}=(x^{5})^{\frac{1}{2}}=x^{\frac{5}{2}}\). Then the function \(f(x)\) can be written as \(f(x)=2x^{\frac{5}{2}}\).

Step2: Apply the power rule of differentiation

The power rule of differentiation states that if \(y = ax^{n}\), where \(a\) is a constant and \(n\) is a real number, then \(y^\prime=anx^{n - 1}\).
For the function \(f(x)=2x^{\frac{5}{2}}\), here \(a = 2\) and \(n=\frac{5}{2}\).
Using the power rule, we have \(f^\prime(x)=2\times\frac{5}{2}x^{\frac{5}{2}-1}\).

Step3: Simplify the expression

Simplify the coefficient and the exponent:
The coefficient \(2\times\frac{5}{2}=5\).
The exponent \(\frac{5}{2}-1=\frac{5 - 2}{2}=\frac{3}{2}\).
So, \(f^\prime(x)=5x^{\frac{3}{2}}\). We can also rewrite \(x^{\frac{3}{2}}\) as \(\sqrt{x^{3}}\) or \(x\sqrt{x}\), but \(5x^{\frac{3}{2}}\) is a simplified form.

Answer:

\(5x^{\frac{3}{2}}\) (or \(5\sqrt{x^{3}}\) or \(5x\sqrt{x}\))