Derivatives of Exponential Functions | Summary and Q&A

TL;DR

This video explains how to find derivatives of exponential functions using the derivative rules for e and other bases.

Key Insights

• ⌛ The derivative of e to the u, where u is a function, is e to the u times the derivative of u.
• ⌛ For bases other than e, the formula for finding the derivative is a to the u times u prime times ln a.
• 📏 Harder problems may require the use of the product rule or quotient rule to find the derivative.

Transcript

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Questions & Answers

Q: What is the derivative of e to the x?

The derivative of e to the x is e to the x times the derivative of x, which is 1.

Q: How do you find the derivative of e to the x cubed plus 8x?

The derivative is e to the x cubed plus 8x times the derivative of the exponent. The derivative of x cubed is 3x squared and the derivative of 8x is 8.

Q: How do you find the derivative of 3 raised to the x?

The derivative of 3 raised to the x is 3 raised to the x times the derivative of x, which is 1, multiplied by ln 3.

Q: What is the derivative of e raised to the sine x?

The derivative is e raised to the sine x times the derivative of sine x, which is cosine x.

Q: How do you find the derivative of x cubed e to the 4x?

Use the product rule. The derivative is x squared e to the 4x plus 3x squared e to the 4x.

Summary & Key Takeaways

• The video begins by introducing the formula for finding the derivative of e to the u, where u is some function. The derivative is e to the u times the derivative of u.

• Examples are shown for finding the derivatives of expressions like e to the 5x plus 3 and e to the x squared.

• The video also covers finding derivatives for bases other than e, using the formula a to the u times u prime times ln a.

• Harder problems involving multiple steps and the product and quotient rules are demonstrated.