More about e

Previously I posted about a few of the remarkable properties of e, the base of natural logarithms. I said that e^x is the function that is the derivative of itself.

Actually, that's not the whole truth. Every function b^x is the derivative of itself, you just have to multiply by a constant. The thing is that the constant is 1 when b = e. As you can see in the code below, we obtain the slope of the curve b^x at x as log_e b * b^x. The constant to multiply by is log_e b, which equals 1 when b = e.

For more details, see Chapter 6 in Strang.

color.list=c( 'red','magenta', 'cyan','thistle4', 'salmon') plot(1:15,xlim=c(0.5,1.8), ylim=c(0,15),type='n') L = 2:6 L2 = c(1,1.5) for (i in 1:length(L)) { b = L[i] col=color.list[i] curve(b**x,lwd=10, add=T,col=col) for (j in 1:length(L2)) { x = L2[j] dx = 0.35 x0 = x - dx x1 = x + dx y = b**x points(x,y,pch=16, col='blue',cex=2) m = log(b)*y # slope is const * y y0 = y - m*dx y1 = y + m*dx lines(c(x0,x1), c(y0,y1),lwd=3, col='blue') } }