Find expert advice and community support for all your questions on IDNLearn.com. Join our knowledgeable community and get detailed, reliable answers to all your questions.
Sagot :
To evaluate [tex]\(\frac{d}{d x} \int_a^x f(t) \, dt\)[/tex] and [tex]\(\frac{d}{d x} \int_a^b f(t) \, dt\)[/tex], let’s analyze them step-by-step.
### 1. [tex]\(\frac{d}{d x} \int_a^x f(t) \, dt\)[/tex]:
For this part, we can use the Fundamental Theorem of Calculus, Part 1, which states that if [tex]\(F(x) = \int_a^x f(t) \, dt\)[/tex], then the derivative [tex]\(F'(x)\)[/tex] is simply the integrand evaluated at [tex]\(x\)[/tex].
[tex]\[ \frac{d}{d x} \int_a^x f(t) \, dt = f(x) \][/tex]
This theorem essentially tells us that the derivative of the integral, with a variable upper limit, of a continuous function [tex]\(f(t)\)[/tex] is the function itself evaluated at the upper limit.
### 2. [tex]\(\frac{d}{d x} \int_a^b f(t) \, dt\)[/tex]:
In this case, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants. Therefore, the integral [tex]\(\int_a^b f(t) \, dt\)[/tex] evaluates to a constant value regardless of [tex]\(x\)[/tex]. The derivative of a constant with respect to [tex]\(x\)[/tex] is zero.
[tex]\[ \frac{d}{d x} \int_a^b f(t) \, dt = 0 \][/tex]
### Summary:
Combining both results, we get:
[tex]\[ \frac{d}{d x} \int_a^x f(t) \, dt = f(x) \][/tex]
[tex]\[ \frac{d}{d x} \int_a^b f(t) \, dt = 0 \][/tex]
These results align with our understanding of calculus and the behavior of integrals with respect to differentiation.
### 1. [tex]\(\frac{d}{d x} \int_a^x f(t) \, dt\)[/tex]:
For this part, we can use the Fundamental Theorem of Calculus, Part 1, which states that if [tex]\(F(x) = \int_a^x f(t) \, dt\)[/tex], then the derivative [tex]\(F'(x)\)[/tex] is simply the integrand evaluated at [tex]\(x\)[/tex].
[tex]\[ \frac{d}{d x} \int_a^x f(t) \, dt = f(x) \][/tex]
This theorem essentially tells us that the derivative of the integral, with a variable upper limit, of a continuous function [tex]\(f(t)\)[/tex] is the function itself evaluated at the upper limit.
### 2. [tex]\(\frac{d}{d x} \int_a^b f(t) \, dt\)[/tex]:
In this case, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants. Therefore, the integral [tex]\(\int_a^b f(t) \, dt\)[/tex] evaluates to a constant value regardless of [tex]\(x\)[/tex]. The derivative of a constant with respect to [tex]\(x\)[/tex] is zero.
[tex]\[ \frac{d}{d x} \int_a^b f(t) \, dt = 0 \][/tex]
### Summary:
Combining both results, we get:
[tex]\[ \frac{d}{d x} \int_a^x f(t) \, dt = f(x) \][/tex]
[tex]\[ \frac{d}{d x} \int_a^b f(t) \, dt = 0 \][/tex]
These results align with our understanding of calculus and the behavior of integrals with respect to differentiation.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.