The exponential function has a very special shape. The slope at each point in domain is equal to the value at that point.

Exponential Function is an extension of a polynomial whose power goes to infinity but the sum still converge. Suppose $f\left(x\right)$ represents the exponential function.
Write $$f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+\cdots }$$ and $$f^{\prime }\left(x\right)={\displaystyle \sum _{n=1}^{\infty }n{a}_n{x}^{n-1}}=a_1+2a_{2}x+3a_3{x}^2+\cdots$$

By the geometrical motivation, the two polynomials must be identical: $$\begin{array}{l}{a}_1=a_0=\frac{a_0}{1!}\\ a_2=\frac{1}{2}{a}_1=\frac{a_0}{2!}\\ a_3=\frac{1}{3}{a}_2=\frac{a_0}{3!}\\ \cdots \\ a_n=\frac{a_0}{n!}\end{array}$$

Hence $$f\left(x\right)=a_0{\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n!}}$$ and we denote the kernel ${\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n!}}$ as $\exp \left(x\right)$.