Fundamental theorem of calculus (part 1) example#2

TL;DR

Learn how to find the derivative of a function using the Fundamental Theorem of Calculus.

Transcript

another ftc one question we are going to find the derivative of the function this is defined in terms of integral integral from 1 minus 3x to 1 and we have a crazy expression instead of the integral if this one says don't worry about how to integrate this because derivative will cancel out integral but before we can use ftc one we have to make sure... Read More

Key Insights

• 🍉 Adjusting the limits of integration is crucial when finding the derivative of a function defined in terms of an integral.
• 😑 The chain rule must be applied to the expression with x before finding the derivative.
• 😑 The Fundamental Theorem of Calculus allows us to find the derivative of a function by plugging in the expression with x and multiplying it by the derivative of that expression.

Questions & Answers

Q: How do you adjust the limits of integration when finding the derivative of a function defined in terms of an integral?

To adjust the limits, make sure the number is on the bottom and the expression with x is on the top. If the order is switched, rewrite the integral accordingly and include a negative sign.

Q: What is the next step after adjusting the limits of integration?

The next step is to substitute the expression with x into the function and find the derivative of that part. Multiply the result by the derivative of the outer part (chain rule) and simplify if needed.

Q: How does the Fundamental Theorem of Calculus come into play?

The Fundamental Theorem of Calculus allows us to find the derivative of a function defined in terms of an integral. By plugging in the expression with x and multiplying it by the derivative of that expression, we can obtain the derivative of the function.

Q: Why is it important to change the sign when switching the limits of integration?

Changing the sign is necessary because when switching the limits, it affects the orientation of the integral. This ensures that the negatives cancel out correctly in the derivative calculation.

Summary & Key Takeaways

• The video teaches how to find the derivative of a function defined in terms of an integral.

• The limits of integration must be adjusted to ensure the number is on the bottom and the expression with x is on the top.

• By applying the Fundamental Theorem of Calculus, the derivative of the function can be found.