Differentiation and Integration with Bases other than e  Full Calculus Tutorial  Summary and Q&A
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.
Key Insights
 β Calculus involves important formulas for derivatives and integrals.
 βοΈ The derivative formula for a to the x is a to the x multiplied by the natural logarithm of a.
 ποΈ The integral formula for a to the x is a to the x divided by the natural logarithm of a, plus a constant.
 βΊοΈ The derivative formula for logs with bases other than e is 1 over x times 1 over the natural logarithm of the base.
 β These formulas can be used to find derivatives and integrals of various functions.
 β It is important to understand the proofs of these formulas to fully grasp their concepts.
 β Memorizing these formulas can be helpful in solving calculus problems efficiently.
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
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.