Problem 30

Question

Differentiate $$ f(x)=(a-x)(a+x) $$ with respect to $x$. Assume that $a$ is a positive constant.

Step-by-Step Solution

Verified

Answer

The derivative is $f'(x) = -2x$.

1Step 1: Identify the Function

The function given is a product of two terms, i.e., $f(x) = (a-x)(a+x)$. We will need to use the product rule for differentiation.

2Step 2: Simplify the Expression

First, recognize that the expression $(a-x)(a+x)$ is a difference of squares. Therefore, simplify it to $f(x) = a^2 - x^2$.

3Step 3: Differentiate Using Power Rule

Now differentiate the simplified expression $f(x) = a^2 - x^2$. The derivative of $a^2$ is 0 (since $a$ is a constant) and the derivative of $-x^2$ is $-2x$. Therefore, $f'(x) = -2x$.

Key Concepts

DifferentiationProduct RulePower Rule

Differentiation

Differentiation is a fundamental concept in calculus, focused on finding the derivative of a function. When we differentiate, we seek the rate at which a quantity changes. It's like asking how fast something is happening at any point. Imagine you're driving, and you want to know your speed at a particular moment. Differentiation gives you that speed, illustrating how your position changes over time.

In our exercise, we need to differentiate the function $f(x) = (a-x)(a+x)$. This involves finding $f'(x)$, the derivative with respect to $x$. This approach will help uncover how the function behaves as $x$ changes. Remember, the function involves simple algebraic terms, hence we are essentially looking to find how changes in $x$ affect the function value. Differentiation becomes the tool that helps us determine this effect.

Product Rule

The Product Rule in calculus is a specialized technique used to differentiate functions that are the product of two or more terms. This rule comes in handy when we cannot simplify the function easily, so we apply a logical rule to find the derivative of the product.

The rule states: If you have a function $ f(x) = u(x) \cdot v(x) $, then its derivative, $ f'(x) $, is given by:

$ (u(x) \cdot v(x))' = u'(x) \cdot v(x) + u(x) \cdot v'(x) $

In our exercise, the original problem $f(x) = (a-x)(a+x)$ initially looks like an ideal candidate for the Product Rule since it's a product of two expressions. However, by simplifying it, we avoided the need for this rule and made it easier to differentiate. Understanding when to use the Product Rule versus simplifying first can greatly streamline solving these problems.

Power Rule

The Power Rule is a simple yet powerful technique in differentiation. It's used to find the derivative of functions that involve powers, usually polynomials. The rule says: For any function of the form $x^n$, the derivative is $nx^{n-1}$.

In our given function $f(x) = a^2 - x^2$, we apply the Power Rule to $-x^2$. Differentiating $x^2$ gives us $2x$; since we want the derivative of $-x^2$, it becomes $-2x$. The constant $a^2$ simplifies to $0$ because differentiation wipes out constants—it calculates the rate of change, and constants don't change!

Remember, the Power Rule is efficient and widely used in calculus, especially for polynomial expressions. It's like the toolkit essential for any calculus enthusiast.

Problem 30

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