Problem 39 - Calculus of a Single Variable

Question

In Exercises 39–52, find the derivative of the function. $$ f(x)=x^{2}+5-3 x^{-2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^{2} + 5 - 3x^{-2}\) is \(f'(x) = 2x - 6x^{-3}\).

Step 1: Identify the terms

The given function \(f(x) = x^{2} + 5 - 3x^{-2}\) has three terms. They are \(x^{2}\), 5 and \(3x^{-2}\).

Step 2: Differentiate using power rule

Apply the power rule to each term: \n\nThe derivative of the first term \(x^{2}\) is \(2x^{1} = 2x\). \n\nThe derivative of the constant term 5 is 0 as the derivative of a constant value is zero.\n\nThe derivative of the third term \(3x^{-2} = -3 \* 2 \* x^{-3} = -6x^{-3}\).

Step 3: Combine the derivatives

Determine the derivative of the function by combining the derivatives of each term. It is given by: \(f'(x) = 2x - 0 - 6x^{-3} = 2x - 6x^{-3}\).

Key Concepts

The Power Rule in Differentiation

When learning calculus, one of the most fundamental tools you will encounter is the power rule for differentiation. This rule greatly simplifies the process of finding the derivative of polynomials and any other functions where the variable is raised to a power. To apply the power rule, we take any term in the form of \(x^n\) and differentiate by bringing down the exponent as a multiplier and subtracting one from the exponent.

For example, when differentiating \(x^2\), you multiply by the current exponent, 2, and then decrease that exponent by 1, resulting in \(2x^1\), which simplifies to \(2x\). This process is repeated for each term in the polynomial that has a variable raised to a power. It's crucial to remember, regardless of whether the exponent is positive or negative (which we'll address later), the power rule applies all the same. Short and sweet—the power rule is an invaluable shortcut in calculus for expediting the differentiation of polynomial terms.

Derivative of a Constant

When differentiating functions, it's important to know that the derivative of a constant is always zero. If you encounter a standalone number in your function, such as 5 in \(f(x) = x^2 + 5 - 3x^{-2}\), this part of the function does not change as x changes; therefore, it has no rate of change.

In the context of our example, when you apply differentiation to the constant term, the result is:\(\frac{d}{dx} (5) = 0\). This simplifies your work, because any term in the function that's a constant can be immediately set aside as contributing nothing to the overall derivative. Remember this principle to avoid unnecessary calculations and streamline the process of finding derivatives of more complex functions.

Dealing with Negative Exponents

Negative exponents in a function can present an extra step in differentiation, but with a solid understanding of the rules, they pose little trouble. A term like \(-3x^{-2}\) can be intimidating at first, but it's simply interpreted as \(-3/x^2\). When differentiating this term using the power rule, treat the negative exponent as you would any other: multiply by the exponent and subtract one from it.

Following this method, the derivative becomes:\(-3 \cdot -2x^{-2-1} = 6x^{-3}\), or, written with positive exponents, \(\frac{6}{x^3}\). Understanding how to handle negative exponents in differentiation is critical for accurately determining the derivatives of functions with such terms. It's a straightforward application of the power rule, with the sign adjustment being the only additional consideration.