Problem 52 - Calculus and its applications

Question

Find \(y^{\prime}\) $$ \text { If } y=\frac{4}{x^{2}}, \text { find }\left.\frac{d y}{d x}\right|_{x=-2} $$

Step-by-Step Solution

Answer

The derivative \( \left.\frac{dy}{dx}\right|_{x=-2} \) is 1.

Step 1: Understand the function

We are given the function \( y = \frac{4}{x^2} \). This is a rational function where the numerator is a constant and the denominator is \( x^2 \).

Step 2: Apply the power rule

The function \( y = \frac{4}{x^2} \) can be rewritten using negative exponents as \( y = 4x^{-2} \). To differentiate \( y \), we will use the power rule, which states that \( \frac{d}{dx} x^n = nx^{n-1} \).

Step 3: Differentiate the function

Applying the power rule to \( y = 4x^{-2} \), we get \( y' = 4 \times (-2) \times x^{-3} = -8x^{-3} \).

Step 4: Evaluate the derivative at \( x = -2 \)

Now that we have \( y' = -8x^{-3} \), substitute \( x = -2 \) into the derivative: \( y'(-2) = -8(-2)^{-3} \).

Step 5: Simplify the expression

Calculate \( (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} \). Thus, \( y'(-2) = -8 \times \frac{1}{-8} = 1 \).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus used to find the derivative of functions of the form \( x^n \). It's like a magic tool that simplifies the process of differentiation. The rule says that the derivative of \( x^n \) is \( nx^{n-1} \). This means you multiply the original exponent by the coefficient and then subtract one from the exponent.

• For example, the derivative of \( x^3 \) would be \( 3x^2 \).
• If you had \( 5x^4 \), applying the power rule gives \( 20x^3 \) (since \( 4 \times 5 = 20 \)).

Using the power rule simplifies tasks of finding derivatives, especially in polynomial expressions. It's essential for those dealing with any course involving calculus. During calculus exercises, identifying and applying the right rule, like the power rule, can be a game-changer in solving differentiation problems.
In the solution to our exercise, we used the power rule on \( 4x^{-2} \), transforming it into \(-8x^{-3} \). This transformation is at the core of solving our problem.

Rational Functions

Rational functions are like mathematical fractions. They involve two polynomials and one is in the denominator. Think of them as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The function we are dealing with in our exercise, \( \frac{4}{x^2} \), is a perfect example.

Rational functions can have properties like vertical asymptotes, where the function tends toward infinity at a specific point.

They can also have horizontal or oblique asymptotes, describing behavior as \( x \) goes to infinity or negative infinity.

Dealing with rational functions often involves understanding how the variables affect the function. For differentiation, rational functions can often be simplified by rewriting them to use the power rule. In our problem, we did this by rewriting \( \frac{4}{x^2} \) as \( 4x^{-2} \). This made the differentiation process straightforward by allowing us to apply the power rule easily.

Negative Exponents

Negative exponents are a way to express division involving powers. Instead of writing \( \frac{1}{x^n} \), you can use \( x^{-n} \). It's a convenient notational shortcut that often simplifies calculations.

• For instance, \( x^{-3} \) is equivalent to \( \frac{1}{x^3} \).
• Similarly, \( 4x^{-2} \) represents \( \frac{4}{x^2} \).

When dealing with negative exponents, it's important to understand their impact on calculus. They allow the rewriting of functions in a way that simplifies differentiation. For example, in our exercise, converting \( \frac{4}{x^2} \) to \( 4x^{-2} \) let us directly apply the power rule.
Always remember that when you differentiate a function with a negative exponent, the result maintains the negative exponent form unless simplified further. This helps clarify expression outcomes, aiding in accurate manipulation of functions for calculus operations.