Fractional Differentials

Calculus Level 4

$\large \frac{d^{3/2}}{dx^{3/2}}x^2$

Evaluate the above at $x=4\pi$ .

The answer is 8.

1 solution

Kb E
Nov 23, 2017

We have that $\frac{d^a}{dx^a} x^k = \frac{\Gamma(k+1)}{\Gamma(k-a+1)} x^{k-a}$ for $k\geq 0$ Therefore, $\frac{d^{3/2}}{dx^{3/2}} x^2 = \frac{\Gamma(3)}{\Gamma(3/2)} x^{1/2}= \frac{2}{\frac{\sqrt{\pi}}{2}} x^{1/2}= \frac{4}{\sqrt{\pi}} x^{1/2}$ .

At $x = 4\pi$ , this becomes $\frac{4}{\sqrt{\pi}} (4\pi)^{1/2} = \frac{4}{\sqrt{\pi}} 2\sqrt{\pi} = 8$ .

@Kaan Berk Erdogmus

@Digvijay Singh

Is it true, that $\frac{d^a}{dx^a} x^k = \frac{\Gamma(k+1)}{\Gamma(k-a+1)} x^{k-a} = \frac{k!}{(k-a)!} x^{k-a}$ (for $k\geq 0$ )?

I am just asking, because I want to know whether you can phrase the statement above using factorials.

Ron Lauterbach - 3 years, 6 months ago

Yes, for positive integer values of k and a and k-a>0, you could write it in terms of factorials. Of course, if you defined the factorial in terms of the gamma function -which some calculators do- then your statement would be true for all reals k >=0.

kb e - 3 years, 6 months ago

Fractional Differentials

The answer is 8.

1 solution

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