## Calculus, 10th Edition (Anton)

The derivative, or slope of the tangent line, of $f(x)=b^{x}$ is $f'(x)=b^{x}ln(b)$. If the slope of tangent line at $x=0$ is $1$, we know that $$f'(0)=1$$ Replacing $f'(0)$ with the derivative at $x=0$ yields $$b^{0}ln(b)=1$$ $$ln(b)=1$$ Solving for $b$ by raising each side of the equation to the $e$ yields $b=e$. An exponential function with base $e$ is the natural exponential function.