Finding derivative with fundamental theorem of calculus: x is on both bounds | Khan Academy

TL;DR

The video explains how to take the derivative of a definite integral using the fundamental theorem of calculus.

Transcript

So let's see if we can take the derivative of this expression right over here, if we can find capital F prime of x. And once again, it looks like you might be able to use the fundamental theorem of calculus. A big giveaway is that you're taking the derivative of a definite integral that gives you a function of x. But here I have x on both the upper... Read More

Key Insights

• 🍳 Taking the derivative of a definite integral requires breaking down the integral into two separate integrals using a constant, c.
• 👻 The fundamental theorem of calculus allows us to evaluate these separate integrals.
• 📈 Graphing the function within the integral can help visualize the process and determine the appropriate bounds for the integrals.
• 😑 The resulting derivative can be simplified by combining like terms and rearranging the expression.
• 🆘 Introducing the constant, c, helps in applying the fundamental theorem of calculus and simplifying the overall calculation.
• 🍉 The derivative of the first integral involves a negative term, while the derivative of the second integral has a positive term.
• 🥡 The process of taking the derivative of a definite integral can be applied in various calculus problems.

Questions & Answers

Q: How can the fundamental theorem of calculus be applied to find the derivative of a definite integral?

In the case where the upper and lower bounds of the integral both contain x, the integral can be split into two separate integrals by introducing a constant, c. This allows us to rewrite the expression in a form that can be evaluated using the fundamental theorem of calculus.

Q: What is the purpose of introducing the constant, c, to split the integral into two separate integrals?

Introducing the constant allows us to break down the overall area under the curve into two distinct areas. This makes it easier to evaluate the integral and apply the fundamental theorem of calculus.

Q: How does the derivative of the first integral, from c to x, differ from the derivative of the second integral, from c to x squared?

The derivative of the first integral is negative cosine(x)/x, while the derivative of the second integral is 2cos(x^2)/x. The first integral contributes a negative term to the final derivative, while the second integral adds a positive term.

Q: Can the result of the derivative be further simplified?

Yes, the final result can be simplified to (-x)(2cos(x^2) - cos(x)). By combining like terms and rearranging, we can express the derivative in a more concise form.

Summary & Key Takeaways

• The video discusses the process of taking the derivative of a definite integral and how it relates to the fundamental theorem of calculus.

• By graphing the function within the integral, it becomes apparent that the integral can be split into two separate integrals using a constant, c.

• The final result of the derivative can be simplified to -x(2cos(x^2) - cos(x)).