Use the Limit Definition to Find the Derivative e^x

Problem

d(e^x)/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)=e^x into the limit definition.

d(e^x)/d(x)=(lim_h→0)((e^(x+h)−e^x)/h)

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

d(e^x)/d(x)=(lim_h→0)((e^x*e^h−e^x)/h)

4. Factor out the common term e^x from the numerator.

d(e^x)/d(x)=(lim_h→0)((e^x*(e^h−1))/h)

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

d(e^x)/d(x)=e^x*(lim_h→0)((e^h−1)/h)

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

d(e^x)/d(x)=e^x⋅1

Final Answer

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

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