We can make our own ideas when we want to understand the facts of mathematics and physics:
 What they are about.
 Why they are true.
 How we could discover them.
Contents

Discover the exponential series
We differentiate and integrate zero to discover two functions of x that are the same as their derivatives: Zero, and the exponential series.

Discover the hyperbolic sine and hyperbolic cosine series
We integrate one and x to discover two functions of x that are the same as their second derivatives: The hyperbolic sine and hyperbolic cosine series.

Discover that the derivative of e^{x} is e^{x}
We calculate four values of a series to discover that it is the same function of x as e^{x}. The derivative of the series is also the same as e^{x}.

Why the derivative of e^{x} is e^{x}
We differentiate the exponential series, which is the same function of x as e^{x}. The derivative is the exponential series.

Why the derivative of sin(x) is cos(x)
We differentiate the sine series, which is the same function of x as sin(x). The derivative is the cosine series, cos(x).
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More to come
 Discover the sine and cosine series
 Why the derivative of sinh(x) is cosh(x)
 Why the derivative of cosh(x) is sinh(x)
 Why the second derivative of sinh(x) is sinh(x)
 Why the second derivative of cosh(x) is cosh(x)
 Why the second derivative of sin(x) is −sin(x)
 Why the second derivative of cos(x) is −cos(x)
 Why the derivative of ln(x) is 1∕x
 Why energy is conserved
 Why angular momentum is conserved
 color blower