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The derivative of a function gives us the rate at which the function is changing at any given point. It is an essential tool in calculus that provides insights into the behavior of a function. For the function given by the integral \( F(x) = \int_{2}^{x} \frac{1}{\ln t} \ dt \), the Fundamental Theorem of Calculus states that its derivative \( F'(x) \) is simply the integral's integrand evaluated at \( x \). Here, this means \( F'(x) = \frac{1}{\ln x} \).

This derivative, \( F'(x) = \frac{1}{\ln x} \), helps us quickly determine how the function is behaving. If \( F'(x) > 0 \), the function \( F(x) \) is increasing at that point. Conversely, if \( F'(x) < 0 \), the function is decreasing.

• The derivative tells us if the function is growing or shrinking in value.
• By analyzing \( F'(x) \), we gain insight into how \( F(x) \) behaves over its domain.
• It reflects the local behavior of the function around any point \( x \).

Concavity of a function describes the direction a curve bends, letting us know if it's bulging upwards or downwards. To explore this, we need to find the second derivative. For the function \( F(x) = \int_{2}^{x} \frac{1}{\ln t} \ dt \), the second derivative \( F''(x) \) is found by further differentiating \( F'(x) = \frac{1}{\ln x} \).

Using the chain rule, \( F''(x) = -\frac{1}{(\ln x)^2 \cdot x} \). The negative sign in front indicates that \( F''(x) < 0 \) for all \( x \geq 2 \). This tells us:

• The graph of \( F(x) \) is concave down across its entire domain.
• A concave-down graph curves downwards, like an upside-down bowl.
• This property can help in sketching the graph since every part of \( F(x) \) will have this downward curvature.

Knowing when a function is increasing or decreasing is crucial for understanding its overall behavior. A function is increasing in intervals where its derivative \( F'(x) \) is positive and decreasing where \( F'(x) \) is negative. For \( F(x) = \int_{2}^{x} \frac{1}{\ln t} \ dt \), the critical factor is \( F'(x) = \frac{1}{\ln x} \).

For \( x > e \), \( \ln x > 0 \) which makes \( F'(x) > 0 \). Therefore, \( F(x) \) is increasing in the interval \((e, \infty)\). For \( 2 \leq x < e \), however, we have \( \ln x < 0 \) and thus \( F'(x) < 0 \), indicating that \( F(x) \) is decreasing in this interval.

• Identifying intervals of increase and decrease helps in predicting long-term behavior.
• The intervals of increase and decrease directly affect how the graph of \( F(x) \) is sketched.
• This information can also confirm or predict specific features such as local maxima or minima.