[MATH DERiVATiVE]: Why is d/dx(e^x) = e^x

[SiRaLeX]

d/dx(0.5^x) = -0.693147×0.5^x

d/dx(1^x) = 0

d/dx(1.5^x) = 0.405465×1.5^x

d/dx(2^x) = 2^x log(2)

 

d/dx(2.71^x) = 0.996949×2.71^x

d/dx(e^x) = e^x

d/dx(2.72^x) = 1.00063×2.72^x

 

d/dx(3.14^x) = 1.14422×3.14^x

d/dx(pi^x) = pi^x log(pi)

 

 

Explain, anyone? 

 

 

[Jacko]

d/dx(0.5^x) = -0.693147×0.5^x

d/dx(1^x) = 0

d/dx(1.5^x) = 0.405465×1.5^x

d/dx(2^x) = 2^x log(2)

 

d/dx(2.71^x) = 0.996949×2.71^x

d/dx(e^x) = e^x

d/dx(2.72^x) = 1.00063×2.72^x

 

d/dx(3.14^x) = 1.14422×3.14^x

d/dx(pi^x) = pi^x log(pi)

 

 

Explain, anyone? 

 

Magic.

[Nyerguds]

Man, that shit's been ages. You do your homework yourself, lol.

[GiftS]

d/dx(0.5^x) = -0.693147×0.5^x

d/dx(1^x) = 0

d/dx(1.5^x) = 0.405465×1.5^x

d/dx(2^x) = 2^x log(2)

 

d/dx(2.71^x) = 0.996949×2.71^x

d/dx(e^x) = e^x

d/dx(2.72^x) = 1.00063×2.72^x

 

d/dx(3.14^x) = 1.14422×3.14^x

d/dx(pi^x) = pi^x log(pi)

 

 

Explain, anyone? 

 

Do it by definition if it confuses you... 

 

[Strategst]

The derivative of a function gives its instantaneous rate of change, or slope, at any given point. The exponential function just so happens to have the same value as its slope at every point. The simplest example is exp^0=1, where the slope of exp^x at x=0 is also 1. Continue inductively from there.

[SiRaLeX]

Strategst, I know what a derivative is.

 

 

Magic.

Do it by definition if it confuses you... 

 

 

These are the best answers. Except, I'm not sure I can simplify the limit without L'H and knowing that the derivative of e^x is e^x. 

 

 

[Plokite_Wolf]

These are the best answers. Except, I'm not sure I can simplify the limit without L'H and knowing that the derivative of e^x is e^x. 

There is something called a Google search, which gives the answer (without the L'Hospital rule)...

https://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/e_to_the_x

[SiRaLeX]

Thank you, Plokite_Wolf! It is all over Google, indeed. But your link is exactly what I was looking for!