Question

In Exercises, evaluate the derivative of the function at the indicated points on the graph. $$ f(x)=x+\frac{32}{x^{2}} $$

Short answer

The derivative of the function \( f(x) = x + \frac{32}{x^2} \) is \( f'(x) = 1 - \frac{64} {x^3} \).

Step-by-step solution

Recognize the Format of the Function

The given function \( f(x) = x + \frac{32}{x^2} \) can be rewritten as \( f(x) = x + 32x^{-2} \). This format makes it easier to apply the power rule.

Apply the Power Rule

The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Apply the power rule to both terms to get \( f'(x) = 1.0 + 32(-2)x^{-3} \). Simplify to get \( f'(x) = 1 - 64x^{-3} \).

Replace the \( x^{-3} \) Term

The derivative can be further simplified by replacing the term with negative exponent to its reciprocal form. Thus, \( f'(x) = 1 - \frac{64} {x^3} \). This is the derivative of the function. To find the derivative at specific points on the graph, replace \( x \) with the desired values.

Key concepts

Power Rule

When working with derivatives, one of the most fundamental tools is the power rule. It's a shortcut that makes finding the rate of change for terms like \(x^n\) a breeze. The rule says that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). In practice, you simply multiply the exponent by the coefficient and then subtract one from the exponent.

For example, if we have a function \(g(x)=x^5\), applying the power rule gives us \(g'(x)=5x^4\). The power exponentially reduces by one, and it’s applied to any variable raised to a power, saving time and reducing complexity in calculations. It's especially useful in simplifying derivatives of polynomial functions, making this tool one of the first to reach for when confronted with these types of expressions.

Negative Exponent

Working with negative exponents might initially seem challenging, but with a proper understanding, you can turn it into a simple task. A negative exponent indicates that the number or variable is on the bottom of a fraction. To simplify an expression with a negative exponent, you turn it into a positive exponent by taking the reciprocal of the base, effectively 'flipping' the fraction.

For example, \(x^{-2}\) is equivalent to \(\frac{1}{x^2}\). When calculating derivatives, negative exponents will appear as you apply the power rule to terms that are originally fractions, as was seen in the step-by-step solution. Knowing how to deal with them not only simplifies the differentiation process but also prepares you for working with complex algebraic expressions and calculus problems.

Function Simplification

Function simplification is a mathematical practice that helps in reducing expressions to their most basic form. This process is particularly important in calculus, where simplifying a function's derivative can make it more comprehensible and much easier to evaluate at specific points.

In the context of derivatives, after applying rules like the power rule, your result might include complex fractions, negative exponents, or factors that can cancel out. By simplifying these, you streamline the function for further applications, be it evaluation for specific values of \(x\), graphing, or integrating. A part of simplification could involve transforming negative exponents into positive ones by moving them to the denominator, as was done in the solution provided. It's a crucial part of calculus that not only enhances understanding but also allows for more accurate and efficient mathematical operations.