Nib asked
Why is the derivative of e^x e^x?
The slope of a^x is a^x times a fixed number, the slope at zero; e is the base that makes that number exactly one, so its slope equals its height everywhere.
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1slope = height
For y = e^x, the height at x = 1 is e, about 2.718, and so is the slope of the tangent there. At x = 2 both are about 7.389. The steepness of the curve always matches its height.
For e^x, slope equals height at every point.
a common mix-upIt is not a coincidence or a rule to memorize: e is built so that the slope equals the height.
2factor it out
Take any base a. The slope of a^x is the limit of (a^(x+h) - a^x)/h as h shrinks to zero. Since a^(x+h) = a^x a^h, factor out a^x to get a^x times (a^h - 1)/h.
The second factor does not depend on x at all: it is one fixed number for each base a.
d/dx a^x = a^x times a constant that depends only on a.
3what is e?
As h shrinks, (2^h - 1)/h approaches 0.693 and (3^h - 1)/h approaches 1.099. These are the slopes of 2^x and 3^x at x = 0. Some base between 2 and 3 has slope exactly 1 there: that base is called e, about 2.71828.
e is the base whose exponential has slope 1 at x = 0.
4slope = height
With base e the leftover factor is exactly 1, so d/dx e^x = e^x. Multiplying by a constant C keeps the property, and any function with f' = f is of the form C e^x.
d/dx e^x = e^x, because e is the base with slope 1 at zero.
What is the derivative of 5e^x?
Show the answer
5e^x, since a constant multiple keeps the property f' = f.
5the whole idea
Compare 2^x, e^x and 3^x. Each has slope k times its height, with k = 0.693, 1 and 1.099 respectively. e is the unique base where k = 1, so e^x is its own derivative.
Slope = k x height; e is the base with k = 1.
The derivative of e to the x is itself because e was chosen as the one base whose slope at zero is exactly one.