lpetrich
Contributor
Michael Penn runs a nice YouTube channel: (3) Michael Penn - YouTube
I recall him making a video about deriving trigonometric functions' properties from their differential equations, and I will redo that reasoning here.
First, a simpler version of that reasoning, with the exponential function:
y' = y
For independent variable x, call the solution e(x). Since the integration constant multiplies it, let us take e(0) = 1 for definiteness.
It is easy to show that a shifted solution e(x+y) is also a solution, and that it must differ by a constant:
e(x+y) = A*e(x) = B*e(y) = C*e(x)*e(y)
From e(0) = 1, e(x+y) = e(x)*e(y) -- the exponential function
Let's consider its inverse function, ie: e(ie(x)) = ie(e(x)) = x. Take the derivative:
e'(ie(x)) * ie'(x) = 1 -- e(ie(x) * ie'(x) = 1
ie'(e(x)) * e'(x) = 1 -- ie'(e(x)) * e(x) = 1
giving ie'(x) = 1/x -- ie(x) is the natural-logarithm function of x
I recall him making a video about deriving trigonometric functions' properties from their differential equations, and I will redo that reasoning here.
First, a simpler version of that reasoning, with the exponential function:
y' = y
For independent variable x, call the solution e(x). Since the integration constant multiplies it, let us take e(0) = 1 for definiteness.
It is easy to show that a shifted solution e(x+y) is also a solution, and that it must differ by a constant:
e(x+y) = A*e(x) = B*e(y) = C*e(x)*e(y)
From e(0) = 1, e(x+y) = e(x)*e(y) -- the exponential function
Let's consider its inverse function, ie: e(ie(x)) = ie(e(x)) = x. Take the derivative:
e'(ie(x)) * ie'(x) = 1 -- e(ie(x) * ie'(x) = 1
ie'(e(x)) * e'(x) = 1 -- ie'(e(x)) * e(x) = 1
giving ie'(x) = 1/x -- ie(x) is the natural-logarithm function of x