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.
