$\frac{dx^5}{dx^2},$ the derivative of $x^5$ with respect to $x^2$ 
Let $f(x)=x^5.$ Find its derivative with respect to $x^2$ , i.e., find $\frac{dx^5}{dx^2}.$

We know that $\frac{dx^5}{dx}=5x^4.$
But what should I do when it is needed to take derivative with respect to $x^2 ,x^3$ or other weird things different from classical derivatives?
My approaches:
$1-)$ Say $x^2=a$ , then $x^5 =a^{5/2}$. Hence , find $$\frac{da^{5/2}}{da}=\frac{5}{2}a^{3/2}=\frac{5}{2}x^{3}$$
$2-)$ Find $$\lim_{h\to 0}\frac{f(x^2+h)-f(x^2)}{h}=\lim_{h\to 0}\frac{(x^2+h)^5-(x^2)^5}{h}=5x^8$$
My approaches conflict. Why do they give different results?
Addendum: Extra examples will be appreciated, for example, $dx^7 /dx^3,\;dx^6/dx^5.$

Here is how a Physicist would do it
$$\frac{dx^5}{dx^2}\frac{dx^2}{dx}=\frac{dx^5}{dx}=5x^4$$
Since $\frac{dx^2}{dx}=2x$, we get
$$\frac{dx^5}{dx^2}=\frac{5x^3}{2}$$
