Derivative Proofs

Though there are many different ways to prove the rules for finding a
derivative
, the most common way to set up a proof of these rules is to
go back to the
limit definition
. This way, we can see how the limit definition works for
various functions.

We must remember that mathematics is a succession. It builds on itself, so many
proofs rely on results of other proofs – more specifically, complex proofs of derivatives
rely on knowing basic derivatives. We can also use derivative rules to prove derivatives,
but even those are build off of basic principles in
Calculus
. For the sake of brevity, we won’t go through every proof, but
it is important to know how many of these derivatives were obtained.

It is important to understand that we are not simply “proving a derivative,” but seeing how various rules work for computing the derivative.



Derivative proof of Power Rule



Derivative proofs of ex



Derivative proof of ax


Derivative
proof of lnx


Derivative
proof of sin(x)


Derivative
proof of cos(x)


Derivative
proof of tanx


Derivative
proofs of cotx, secx, and cscx



Derivative proofs of Inverse Trig Functions

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