Problem 35 - Precalculus with Limits

Question

In Exercises 29-42, find the derivative of the function. \(f(x) = \dfrac{1}{x^2}\)

Step-by-Step Solution

Answer

\( \frac{-2}{x^3} \)

Step 1: Rewrite the function

First, rewrite the function \( f(x) = \dfrac{1}{x^2} \) in a form that will allow easy application of the power rule for derivatives. The function can be written as \( f(x) = x^{-2} \)

Step 2: Apply the power rule

Next, apply the power rule to the function \( f(x) = x^{-2} \). The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \), so the derivative of \( f(x) = x^{-2} \) is \( f'(x) = -2x^{-2-1} \)

Step 3: Simplify the derivative

Finally, simplify the derivative, \( f'(x) = -2x^{-2-1} \). This becomes \( f'(x) = -2x^{-3} \) or equivalently \( f'(x) = \dfrac{-2}{x^3} \)

Key Concepts

Power Rule

One of the most fundamental techniques in calculus is the power rule for differentiation. It is used to find the derivative of power functions, where the variable is raised to a constant power. The rule is straightforward: for any function of the form
\( f(x) = x^n \),
where
\( n \) is any real number, the derivative is given by
\( f'(x) = nx^{n-1} \).
This formula greatly simplifies the process of finding derivatives, making it unnecessary to revert to first principles each time. For integers, fractions, and even negative numbers, the power rule applies, and it is the starting point for differentiation of more complex functions.

In the context of the exercise, rewriting \( f(x) = 1/x^2 \) as \( f(x) = x^{-2} \) readies the function for an application of the power rule, transforming the process into a simple multiplication and subtraction operation to find the derivative.

Derivative of Power Functions

Dealing with power functions in calculus involves finding their derivatives using the power rule as previously described. However, recognizing a power function can sometimes be a bit tricky. A power function is any function that can be represented as
\( f(x) = cx^n \),
where
\( c \) is a constant and
\( n \) is a real number exponent.

In our exercise, the function appears initially as a rational expression, but we can view it as a power function with a negative exponent. Often, rewriting the function in the
\( x^n \) format will expose the 'power' nature of the equation. It's essential to recognize power functions because they have predictable behaviors when derived, thus simplifying the process. After identifying the function as a power function, applying the power rule is straightforward and results in
\( f'(x) = -2x^{-3} \).
The negative exponent in the derivative indicates an inverse relationship proportional to the cube of the variable
\( x \).

Simplifying Derivatives

Once we've used the power rule to find the derivative of a power function, the next logical step is to simplify the result for clarity and ease of use, especially in further calculations. Simplification may involve converting negative exponents into fractional forms or combining like terms in more complex scenarios. It's crucial for students to master simplification to express their final answers in the most straightforward form possible.

Referring back to our example, the derivative calculated using the power rule yielded
\( f'(x) = -2x^{-3} \).
To simplify this, we can convert the negative exponent to a fractional form. A negative exponent indicates that the term is actually on the denominator when expressed as a fraction. Thus,
\( x^{-3} \) simplifies to
\( \frac{1}{x^3} \),
and the complete simplified derivative is
\( f'(x) = \frac{-2}{x^3} \).
Learning to simplify derivatives is an invaluable skill, ensuring that solutions are presented in the most understandable and useful form.