Use the Limit Definition to Find the Derivative e^x

Problem

d(ex)/d(x)

Solution

1. State the definition of the derivative using the limit of the difference quotient.

d(ƒ(x))/d(x)=(lim_h→0)((ƒ*(x+h)−ƒ(x))/h)

2. Substitute the function ƒ(x)=ex into the limit definition.

d(ex)/d(x)=(lim_h→0)((e(x+h)−ex)/h)

3. Apply exponent rules to rewrite the term e(x+h) as a product.

d(ex)/d(x)=(lim_h→0)((ex*eh−ex)/h)

4. Factor out the common term ex from the numerator.

d(ex)/d(x)=(lim_h→0)((ex*(eh−1))/h)

5. Move the constant factor ex outside the limit, as it does not depend on h

d(ex)/d(x)=ex*(lim_h→0)((eh−1)/h)

6. Evaluate the limit using the standard calculus limit (lim_h→0)((eh−1)/h)=1

d(ex)/d(x)=ex⋅1

Final Answer

d(ex)/d(x)=ex

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