I was playing around with some simple differential equations earlier and I discovered something cool (at least for me).(adsbygoogle = window.adsbygoogle || []).push({});

Suppose you have [tex]y=sin(x^2) \Rightarrow \frac{dy}{dx}=2xcos(x^2)[/tex]

What if, instead of taking the derivative with respect to [itex]x[/itex], I want to take the derivative with respect to [itex]x^2[/itex], where a simple substitution might help? [tex]u(x)=x^2 \Rightarrow \frac{du}{dx}=2x \Rightarrow du=2xdx[/tex] [tex]y=sin(u) \Rightarrow \frac{dy}{du}=cos(u) \Rightarrow \frac{dy}{dx^2}=cos(x^2)[/tex] If you undo the substitution for both instances of [itex]u[/itex], you arrive at the derivative with respect to [itex]x[/itex] [tex]\frac{dy}{dx^2}=\frac{dy}{2xdx}=cos(u^2) \Rightarrow \frac{dy}{dx}=2xcos(x^2)[/tex]

Now, I understand this result is just an application of the chain rule, but, I decided to use it on integrals. Which turns out great because this is just an application of integration by parts. [tex]\int dy=\int 2xcos(x^2)dx=\int cos(u)du=\int cos(x^2)dx^2[/tex]

Now, my question is, is it possible to apply this to derivatives of higher order? Here is my approach by example. [tex]\frac{d^2y}{dx^2}=6x[/tex] [tex]\int d^2y=\int 6xdx^2,~~~~~~~dx^2=2xdx[/tex] [tex]\int d^2y=\int 6x(2xdx)[/tex]

In analogy to stating [itex]\int dx^2=\int 2xdx[/itex], how would i write [itex]\int d^2y ~~~~[/itex] ?

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# Chain Rule, Differentials "Trick"

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