Problem 100 - Calculus Early Transcendentals

Question

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4}$$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(\frac{2x}{x^2+4}\).

Step 1: Apply the Fundamental Theorem of Calculus

Since we have a derivative of an integral, we can apply the Fundamental Theorem of Calculus (FTC) to simplify the expression. According to the FTC: $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4} = \frac{1}{x^2+4} \cdot \frac{d(x^2)}{dx}$$ Now, we need to find the derivativative of the function \(x^2\) to move to the next step.

Step 2: Find the derivative

We can easily find the derivative of \(x^2\) with respect to \(x\) using the power rule: $$\frac{d(x^2)}{dx} = 2x$$ Now we can substitute this back into our expression.

Step 3: Substitute the derivative and simplify

Finally, we can substitute the derivative of \(x^2\) found in Step 2 back into our expression and simplify: $$\frac{1}{x^2+4} \cdot \frac{d(x^2)}{dx} = \frac{1}{x^2+4} \cdot 2x = \frac{2x}{x^2+4}$$ Our final simplified expression is: $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4} = \frac{2x}{x^2+4}$$

Key Concepts

Derivative

A derivative represents the rate at which a function is changing at any given point.
In simple terms, it's like measuring how fast something is changing. For example, if you're driving, the derivative of your position relative to time is your speed.
For functions, this means calculating the slope of the tangent line at any point. To compute a derivative, you can use different rules and techniques.

• The power rule is one of the most common methods used for finding derivatives.
• Remember, the derivative tells us about the instant rate of change.
• It helps in understanding functions that model real-world situations like motion, growth, and more.

The derivative plays a significant role in calculus and provides a foundation for more complex concepts.

Integral

Integration is essentially the reverse process of differentiation.
While derivatives give you the rate of change, integrals help you find the total accumulation of a quantity.
For instance, if a derivative gives you speed, an integral can help you find the total distance traveled. One of the major applications of integrals is finding areas under curves.
A definite integral calculates the area under a function within specific bounds, essentially summing up tiny bits of area to find a total.
This is done over an interval from one point to another. There are different techniques for integration:

• Substitution method
• Integration by parts
• Partial fraction decomposition

Another critical concept is the Fundamental Theorem of Calculus, which connects derivatives and integrals and facilitates evaluating definite integrals.

Power Rule

The power rule is a straightforward method used to find the derivative of a function of the form \[ f(x) = x^n \].
The rule states that if \[ f(x) = x^n \], then its derivative is \[ f'(x) = n \cdot x^{n-1} \].
This rule simplifies finding the derivative of polynomials, often making calculations quicker and easier.Let's see how it works with a practical example:

• If we have \( x^2 \), applying the power rule results in \( 2x \).
• For \( x^3 \), the derivative would be \( 3x^2 \).

The power rule is especially useful when dealing with simple polynomials, saving time and effort during the computation of derivatives.
It's a powerful tool in the calculus toolkit and is often the first rule students learn when studying derivatives.