homework5: problem 17 (1 point) if $f(x)=3sqrt{x}ln(x)$, find $f(x)$. $f(x)=$ find $f(5)$. $f(5)=$ note: you…

Explanation:

Step1: Apply product - rule

The product - rule states that if (y = u\cdot v), then (y'=u'v + uv'). Let (u = 3\sqrt{x}=3x^{\frac{1}{2}}) and (v=\ln(x)). First, find (u') and (v'). The derivative of (u = 3x^{\frac{1}{2}}) using the power - rule ((x^n)'=nx^{n - 1}) is (u'=\frac{3}{2}x^{-\frac{1}{2}}), and the derivative of (v=\ln(x)) is (v'=\frac{1}{x}).

Step2: Calculate (f'(x))

By the product - rule (f'(x)=u'v + uv'). Substitute (u), (u'), (v), and (v') into the formula: [ \begin{align*} f'(x)&=\frac{3}{2}x^{-\frac{1}{2}}\ln(x)+3x^{\frac{1}{2}}\cdot\frac{1}{x}\ &=\frac{3\ln(x)}{2\sqrt{x}}+\frac{3}{\sqrt{x}}\ &=\frac{3\ln(x)+6}{2\sqrt{x}} \end{align*} ]

Step3: Calculate (f'(5))

Substitute (x = 5) into (f'(x)): [ \begin{align*} f'(5)&=\frac{3\ln(5)+6}{2\sqrt{5}}\ &=\frac{3(\ln(5) + 2)}{2\sqrt{5}} \end{align*} ]

Answer:

(f'(x)=\frac{3\ln(x)+6}{2\sqrt{x}}) (f'(5)=\frac{3(\ln(5)+2)}{2\sqrt{5}})