CCD.DE: The derivative of the natural exponential function. [from Sensible Calculus]
The Derivative of $\exp(x)=e^x$ at $x=0$: $\exp'(0)=1$.

Estimation Argument-"Proof":

Check this visually with the mapping diagram CDD.DEMD.0

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Now that we have found the derivative for $\exp(x)=e^x$ at $ x =0$, it is time to think of this derivatives more generally. In Theorem CCD.DEX we state these results.
Details for the proofs can be found in most calculus texts or in The Sensible Calculus Book Section I.F.2 Derivatives of exponential and logarithmic functions
What is missing in other proofs are visualizations using mapping diagrams. These are provided with the mapping diagram CCD.DEMD.

Theorem CCD.DEX.  $D_x\exp(x) = \exp'(x) =\exp(x)$.

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The Steps refer to the four steps for finding a derivative in CCD.DDN4S.