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While solving integrals using substitution method, we often come across this,

if u = f(x), du = f'(x) dx

I would like to know if there exists a proof for the above equation. The problem is, I am totally dissatisfied by the explanation provided to me in the classes. During the derivatives classes, we were told that in case of (dy)/(dx), it actually means (d/dx)y, where (d/dx) stands as a notation, as opposed to dy being divided by dx.

The way the folks at my classes used substitution in the integral classes was,

let u = f(x)

diff. both sides w.r.t. x

du/dx = f'(x)

thus du = f'(x) dx [Multiplying by dx on both the sides]

But the last step's explanation seems to be too weird to me. How can a part of a notation be canceled off one side? Or is my understanding of the notation in the first place itself is wrong?

Please help me shed some light on the understanding of this dilemma I am facing.

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# Confused at: du = f'(x) dx

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