Question

question 6
let $f(x)=2x^{6}sqrt{x}+\frac{5}{x^{3}sqrt{x}}$.
$f(x)=$

Explanation:

Step1: Rewrite the function

Rewrite $f(x)$ using exponent - form. $\sqrt{x}=x^{\frac{1}{2}}$. So $f(x)=2x^{6}\cdot x^{\frac{1}{2}}+\frac{5}{x^{3}\cdot x^{\frac{1}{2}}}=2x^{6 + \frac{1}{2}}+5x^{-(3+\frac{1}{2})}=2x^{\frac{12 + 1}{2}}+5x^{-\frac{6 + 1}{2}}=2x^{\frac{13}{2}}+5x^{-\frac{7}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.
For the first term $y_1 = 2x^{\frac{13}{2}}$, its derivative $y_1^\prime=2\times\frac{13}{2}x^{\frac{13}{2}-1}=13x^{\frac{13 - 2}{2}}=13x^{\frac{11}{2}}$.
For the second term $y_2 = 5x^{-\frac{7}{2}}$, its derivative $y_2^\prime=5\times(-\frac{7}{2})x^{-\frac{7}{2}-1}=-\frac{35}{2}x^{-\frac{7 + 2}{2}}=-\frac{35}{2}x^{-\frac{9}{2}}$.

Step3: Find the derivative of the function

$f^\prime(x)=y_1^\prime + y_2^\prime=13x^{\frac{11}{2}}-\frac{35}{2}x^{-\frac{9}{2}}$.

Answer:

$13x^{\frac{11}{2}}-\frac{35}{2}x^{-\frac{9}{2}}$