Question

find the given higher - order derivative.
$f(x)=x^{3}-x^{3/4}, f^{(3)}(x)$

Explanation:

Step1: Recall the power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$.

Step2: Differentiate $f^{\prime}(x)=x^{3}-\frac{x^{3}}{4}=\frac{3x^{3}}{4}$ to get $f^{\prime\prime}(x)$

Using the power - rule, if $y=\frac{3x^{3}}{4}$, then $y^\prime=f^{\prime\prime}(x)=\frac{3}{4}\times3x^{2}=\frac{9x^{2}}{4}$

Step3: Differentiate $f^{\prime\prime}(x)$ to get $f^{(3)}(x)$

Applying the power - rule again, if $y = \frac{9x^{2}}{4}$, then $y^\prime=f^{(3)}(x)=\frac{9}{4}\times2x=\frac{9x}{2}$

Answer:

$\frac{9x}{2}$