Proof of the Formula for the Derivative of a^x

TL;DR

The video provides a concise proof that the derivative of a^x is a^x times the natural log of a.

Transcript

in this short video we're going to go through a brief proof of this property from calculus we're going to prove that the derivative of a TV X is a to the x times the natural log of a it's really useful to know how to do this because chances are you might forget this so if you ever forget what the formula is you probably won't forget the proof if yo... Read More

Key Insights

• 🆘 Understanding the proof of the derivative of a^x helps in memorizing the formula.
• 😑 Rewriting a^x as e^(ln(a^x)) is a crucial step in simplifying the expression.
• 👻 The power rule for logarithms allows the exponent to be brought outside the logarithm for differentiation.
• ☺️ Differentiating x times ln(a) results in ln(a) because a is considered a constant coefficient.

Questions & Answers

Q: Why is it important to know the proof of the derivative of a^x?

Knowing the proof helps to understand the concept deeply and allows for the formula to be memorized more easily. It also provides a foundation for further mathematical applications.

Q: How does rewriting a^x as e^(ln(a^x)) help in the proof?

By using the property that e^(ln(x)) = x, we can simplify the expression and cancel out the logarithmic part, making it easier to differentiate.

Q: What is the power rule for logarithms used in the proof?

The power rule states that log(base b)(x^n) = n * log(base b)(x). In the proof, this rule is applied to move the exponent outside the logarithm.

Q: Why does the derivative of x times ln(a) result in ln(a)?

Since a is a constant in this proof, differentiating x times ln(a) treats a as a coefficient. The derivative of x is 1, leaving only ln(a).

Summary & Key Takeaways

• The video presents a proof that the derivative of a^x is a^x times the natural log of a.

• The proof involves rewriting a^x as e^(ln(a^x)), using the power rule for logarithms, and applying the chain rule.

• By simplifying the expression, the video demonstrates that the derivative of a^x is a^x times the natural log of a.