Question

Calculate. $$\frac{d}{d x}\left(\int_{x}^{a} f(t) d t\right)$$

Name: Problem 19, Chapter 5: Calculus: One Variable
Uploaded: 2021-08-21T06:11:06-08:00
Duration: 1 min 4 s
Channel: Adrian Co
Description: Calculate. (d)/(d x)(∫x^a f(t) d t)

   Calculate.
$$\frac{d}{d x}\left(\int_{x}^{a} f(t) d t\right)$$

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 5, Problem 19

Instant Answer

Solved by Expert Adrian Co

08/21/2021

Step 1

This is not the standard form we usually deal with, so we need to change it to the standard form. We can do this by using the property of definite integrals that reverses the limits of integration and changes the sign of the integral. So, we rewrite the integral Show more…

Show all steps

Thanks for your feedback!

Calculate. $$\frac{d}{d x}\left(\int_{x}^{a} f(t) d t\right)$$

Play audio

Feedback

Adrian Co and 65 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Differentiation of Integrals with Variable Limits

In problems where the limits of integration depend on the variable of differentiation, the process involves applying the chain rule to account for the changing limits. Specifically, if the variable is the lower limit—as in the given integral—the derivative of the integral will be negative the value of the integrand evaluated at the variable lower limit, reflecting the reversed orientation of integration.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration. It states that if a function is defined as an integral with one variable limit, then its derivative is equal to the integrand evaluated at that limit. When the variable appears as the upper limit, the derivative is the integrand at that point; if the variable is the lower limit, the derivative includes a negative sign.

Please give Ace some feedback