Problem 25 - Calculus Single Variable

Question

In each of Exercises \(21-28,\) calculate the derivative of \(F(x)\) with respect to \(x\). $$ F(x)=\int_{0}^{x} \tan \left(t^{2}\right) d t $$

Step-by-Step Solution

Answer

The derivative is \( F'(x) = \tan(x^2) \).

Step 1: Identify the function to differentiate

The function given is an integral from 0 to x, specifically: \( F(x) = \int_{0}^{x} \tan(t^2) \, dt \). We need to find \( F'(x) \).

Step 2: Recognize the application of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F'(x) = f(x) \). This applies because the upper limit of the integral matches the variable we are differentiating with respect to.

Step 3: Differentiate the function using the Fundamental Theorem

Applying the Fundamental Theorem of Calculus, we differentiate \( F(x) = \int_{0}^{x} \tan(t^2) \, dt \) with respect to \( x \). Hence, \( F'(x) = \tan(x^2) \).

Step 4: State the derivative

The derivative of the given function \( F(x) \) is \( F'(x) = \tan(x^2) \).

Key Concepts

Derivative Calculation

When we calculate the derivative of a function, we are finding the rate at which the function's value changes with respect to its input. In simpler terms, it's like finding how fast something is moving at any given point in time.
The problem gives us a function defined as an integral: \( F(x) = \int_{0}^{x} \tan(t^2) \, dt \). Our goal is to find \( F'(x) \), the derivative of this function with respect to \( x \).

• Begin by identifying the function \( F(x) \), which in our case is an integral from 0 to \( x \) of another function, \( \tan(t^2) \).
• This process often requires recognizing patterns, like integrals or specific functions, which might be easier to differentiate by a known calculation rule.

Once we know how our function is built, we can then use rules like those provided by the Fundamental Theorem of Calculus to find the derivative quickly and accurately.

Integral Calculus

Integral calculus involves finding the total accumulation of quantities. Think of it as adding up parts to make a whole. In our exercise, we're looking at an integral function, \( F(x) = \int_{0}^{x} \tan(t^2) \, dt \). This integral represents the total accumulation of the area under the curve of \( \tan(t^2) \) from 0 to \( x \).
Here's what to keep in mind:

• An integral can often represent quantities like area, volume, or any total that results from the addition of small parts.
• The variable at the upper limit of the integral, \( x \), plays a crucial role as it defines up to where we're calculating this area.
• When differentiating the integral with respect to \( x \), we essentially calculate how the accumulation totals change as \( x \) changes.

By understanding these concepts, you can see the integral not merely as computation but as a powerful way to understand and model real-world situations.

Differentiation Techniques

Differentiation is a fundamental concept in calculus used to determine how a function changes at any given point. It's not just about finding derivatives, but understanding the collection of techniques to simplify complex problems. In the exercise given, we used the Fundamental Theorem of Calculus to differentiate \( F(x) = \int_{0}^{x} \tan(t^2) \, dt \).
Some important differentiation techniques include:

• Chain Rule: Useful for differentiating compositions of functions.
• Product Rule: Helps differentiate functions that are multiplied together.
• Quotient Rule: Used when differentiating a division of two functions.

In the given case, we observed something unique. Since \( F(x) \) is defined as an integral from 0 to \( x \), the Fundamental Theorem of Calculus states that differentiating \( F(x) \) gives us the function inside the integral evaluated at \( x \).
This knowledge showcases the elegant relationship between differentiation and integration, two core pillars of calculus.