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Fundamental Properties of Differentiation

Michael Green

Michael Green

6 min read

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Study Guide Overview

This study guide covers derivatives of exponential and logarithmic functions. It includes differentiation of , eᵏˣ, aeᵏˣ, , aᵏˣ, ln x, a ln x, and ln kx. The guide provides worked examples, practice questions, and a glossary of key terms like the chain rule and constant multiple rule. Key takeaways emphasize correct application of these rules and properties of logarithms.

Derivative of Exponential and Logarithmic Functions

Table of Contents

  1. Derivative of the Exponential Function
  2. Derivative of a Number Raised to the Power of x
  3. Derivative of the Natural Logarithmic Function
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways

Derivative of the Exponential Function

Differentiating exe^x

The function f(x)=exf(x) = e^x is unique because its rate of change is equal to itself.

f(x)=exf'(x) = e^x

Differentiating ekxe^{kx}

For the function g(x)=ekxg(x) = e^{kx}, its rate of change is proportional to itself.

g(x)=kekxg'(x) = k e^{kx}

This result is due to the chain rule.

Differentiating aekxa e^{kx}

For a function with a constant multiple of the exponential, the differentiation fo...

Question 1 of 12

What is the derivative of f(x)=exf(x) = e^x?

exe^x

1

xex1xe^{x-1}

0