Question

iii. $f(x)=(2x + 1)sqrt{x}$ hint: use product rule

Explanation:

Step1: Identify the functions

Let \(u = 2x + 1\) and \(v=\sqrt{x}=x^{\frac{1}{2}}\).

Step2: Find the derivatives of \(u\) and \(v\)

The derivative of \(u\) with respect to \(x\) is \(u^\prime=\frac{d}{dx}(2x + 1)=2\).
The derivative of \(v\) with respect to \(x\) is \(v^\prime=\frac{d}{dx}(x^{\frac{1}{2}})=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}\).

Step3: Apply the product rule

The product rule states that \((uv)^\prime = u^\prime v+uv^\prime\).
Substitute \(u\), \(u^\prime\), \(v\), and \(v^\prime\) into the formula:
\(f^\prime(x)=(2)\times\sqrt{x}+(2x + 1)\times\frac{1}{2\sqrt{x}}\).

Step4: Simplify the expression

$$
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$$

Answer:

\(f^\prime(x)=\frac{6x + 1}{2\sqrt{x}}\)