What Are the Formulas for Derivatives and Integrals of Exponentials?

TL;DR

The derivative of an exponential function a^x is a^x multiplied by the natural logarithm of a (ln(a)). For logarithmic functions, the derivative of log base a of x is 1/(x * ln(a)). When integrating, the formula for a^x is (1/ln(a)) * a^x + C, highlighting that differentiation involves multiplication by ln(a) while integration entails division.

Transcript

hi everyone in this video we're gonna look at some important formulas so the first formula we're going to look at is the formula for the derivative with respect to X of an exponential function so a to the X so a here is a base so a here is positive and it's not equal to 1 the formula says that this is equal to a to the x times the natural log of a ... Read More

Key Insights

• ❓ Exponential derivative: a^x * ln(a).
• ☺️ Logarithmic derivative: 1/x * 1/ln(a).
• ❓ Integral formula for a^x: 1/ln(a) * a^x + C.
• 🗂️ Derivatives multiply by ln(a), integrals divide by ln(a).
• ❓ Proofs demonstrate the validity of the formulas.
• ⚾ Change of base formula simplifies logarithm derivatives.
• ❓ Understanding the relationship between derivatives and integrals of exponential functions.

Questions & Answers

Q: What is the derivative formula for a^x?

The derivative of a^x is a^x * ln(a), where a is the base.

Q: How is the derivative of log base a of x calculated?

The derivative of log base a of x is 1/x * 1/ln(a), following the chain rule.

Q: What is the integral formula for a^x?

The integral of a^x is 1/ln(a) * a^x + C, where C is the constant of integration.

Q: How do derivatives and integrals of exponential functions relate?

Derivatives involve multiplication by ln(a), while integrals involve division by ln(a).

Summary & Key Takeaways

• Derivative formula for a^x is a^x * ln(a).

• Derivative of log base a of x is 1/x * 1/ln(a).

• Integral formula for a^x is 1/ln(a) * a^x + C.