Differentiation and Integration with Bases other than e - Full Calculus Tutorial

TL;DR

This video explains three important formulas in calculus: the derivative formula for a to the x, the integral formula for a to the x, and the derivative formula for logs with bases other than e.

Transcript

in this video i want to talk about three really really important formulas that are used in calculus so the first one is a formula for the derivative of a to the x so if you take the derivative of a to the x this is simply a to the x natural log of a that's the formula okay so it's a to the x natural log of a let's do a simple example right away so ... Read More

Key Insights

• ❓ Calculus involves important formulas for derivatives and integrals.
• ✖️ The derivative formula for a to the x is a to the x multiplied by the natural logarithm of a.
• 🗂️ The integral formula for a to the x is a to the x divided by the natural logarithm of a, plus a constant.
• ☺️ The derivative formula for logs with bases other than e is 1 over x times 1 over the natural logarithm of the base.
• ❓ These formulas can be used to find derivatives and integrals of various functions.
• ❓ It is important to understand the proofs of these formulas to fully grasp their concepts.
• ❓ Memorizing these formulas can be helpful in solving calculus problems efficiently.

Questions & Answers

Q: What is the formula for the derivative of a to the x?

The formula for the derivative of a to the x is a to the x multiplied by the natural logarithm of a. This means that if we take the derivative of a function like 3 to the x, the result is 3 to the x multiplied by the natural logarithm of 3.

Q: What is the formula for the integral of a to the x?

The formula for the integral of a to the x is a to the x divided by the natural logarithm of a, plus a constant. This means that if we want to find the integral of a function like 4 to the x, the result is 4 to the x divided by the natural logarithm of 4, plus a constant.

Q: What is the formula for the derivative of log base a of x?

The formula for the derivative of log base a of x is 1 over x times 1 over the natural logarithm of a. This means that if we have a function like log base 2 of x, the derivative is 1 over x times 1 over the natural logarithm of 2.

Q: Do these formulas apply to logarithms with base e?

Yes, these formulas also apply to logarithms with base e. For example, if we have the function e to the x, the derivative is e to the x multiplied by the natural logarithm of e, which simplifies to e to the x. Similarly, if we have the integral of e to the x, the result is e to the x divided by the natural logarithm of e, which simplifies to e to the x.

Summary & Key Takeaways

• The first formula discussed is the derivative formula for a to the x, which states that the derivative of a to the x is a to the x multiplied by the natural logarithm of a. An example using 3 to the x is provided.

• The second formula is the integral formula for a to the x, which states that the integral of a to the x is equal to a to the x divided by the natural logarithm of a, plus a constant. An example using 4 to the x is provided.

• The third formula is the derivative formula for logs with bases other than e. It states that the derivative of log base a of x is equal to 1 over x times 1 over the natural logarithm of a. Examples using log base 2 of x and log base 5 of x are provided.