Question

find the derivative of (6sqrt{x}+3sqrt3{x})

Explanation:

Step1: Rewrite the terms

Rewrite $6\sqrt{x}+3\sqrt[3]{x}$ as $6x^{\frac{1}{2}}+3x^{\frac{1}{3}}$.

Step2: Apply power - rule of differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.
For $y_1 = 6x^{\frac{1}{2}}$, $y_1^\prime=6\times\frac{1}{2}x^{\frac{1}{2}-1}=3x^{-\frac{1}{2}}$.
For $y_2 = 3x^{\frac{1}{3}}$, $y_2^\prime=3\times\frac{1}{3}x^{\frac{1}{3}-1}=x^{-\frac{2}{3}}$.

Step3: Combine the derivatives

The derivative of $y = 6x^{\frac{1}{2}}+3x^{\frac{1}{3}}$ is $y^\prime=3x^{-\frac{1}{2}}+x^{-\frac{2}{3}}$.

Answer:

$3x^{-\frac{1}{2}}+x^{-\frac{2}{3}}$