Double-Dog-Derivatives

What happens when the derivative of a function is itself?

As you may already know, a derivative is just a fancy little term for slopes of tangent lines. Or slopes of points, for people who view it differently. Anyway...

There are certain functions whose derivative is itself. What are some of these?

~~~~~~~~~

"e^x" is the only exponential function that has this property. This is just one of the many amazing applications of Euler's number.

"0" is another, but anybody could figure that one out.

"sin(x)" and "cos(x)" have their derivatives every fourth time.

"cosh(x)" and "sinh(x)" also have their own derivatives.

Just a note-- for "log x", the second derivative approaches the original when the base approaches infinity.

No polynomial function works, according to this theorem:
For any polynomial function "f(x)=c*x^n", its derivative is "c*n*x^(n-1)". So even if the coefficient is the same, the degree is offset by one.

~~~~~~~~~~

Sorry for the short post. I started the post, then kind of realized that I didn't really have much to work with...
Hang in there, people! I have final exams coming up, so I might not post much. Just wait, OK?

Stay coolio,
John