Problem 31 - Brief Applied Calculus

Question

31-38. Find the indicated derivatives. If \(f(x)=x^{5}\), find \(f^{\prime}(-2)\).

Step-by-Step Solution

Answer

The derivative \( f'(-2) = 80 \).

Step 1: Recall the Derivative Rule for Powers

To take the derivative of a function of the form \( f(x) = x^n \), recall that the derivative is given by \( f'(x) = n \cdot x^{n-1} \). In this problem, \( n = 5 \).

Step 2: Apply the Power Rule

Using \( f(x) = x^5 \), apply the power rule: \( f'(x) = 5 \cdot x^{4} \). This gives the derivative \( f'(x) = 5x^4 \).

Step 3: Substitute -2 into the Derivative

Now that we have the derivative \( f'(x) = 5x^4 \), substitute \( x = -2 \) into this equation: \( f'(-2) = 5(-2)^{4} \).

Step 4: Simplify the Expression

Calculate \((-2)^4 = 16\). Then multiply by 5: \( 5 \times 16 = 80 \). Therefore, \( f'(-2) = 80 \).

Key Concepts

Power Rule

Understanding the Power Rule is essential when dealing with calculus derivatives. This rule helps us find the derivative of a function that involves a power of a variable. When you see a function expressed as \( f(x) = x^n \), where \( n \) is any real number, the Power Rule states that its derivative, \( f'(x) \), is calculated as \( f'(x) = n \cdot x^{n-1} \). This means you bring down the exponent as a coefficient, then subtract one from the exponent.For example, if your function is \( x^5 \), applying the Power Rule gives you the derivative \( f'(x) = 5 \cdot x^{4} \). It's like peeling a layer off the power; you multiply by the current power and decrease the power by one. This straightforward method is a powerful tool in calculus.

Function Differentiation

Differentiation is the process of finding the derivative of a function, and it tells us how a function changes at any point. The derivative helps us understand the slope or rate of change of a function with respect to an independent variable. When differentiating a simple polynomial function like \( f(x) = x^5 \), we use the Power Rule, as discussed earlier, to efficiently find the derivative.By differentiating \( f(x) = x^5 \), we find \( f'(x) = 5x^4 \). This result shows us the behavior of the function's slope: as \( x \) increases or decreases, the slope changes according to the new power term. Differentiation is key in calculus because it gives insight into the increasing or decreasing behavior of functions and helps in solving complex mathematical problems.

Substitution in Derivatives

Substitution is a crucial step often used after differentiating a function, especially when we want to evaluate the derivative at a specific point. Once you have the derivative expression, substituting a value for \( x \) allows you to find the exact rate of change at that point.In our example, after finding the derivative \( f'(x) = 5x^4 \), we substitute \( x = -2 \) to find \( f'(-2) \). Calculating further, we find \((-2)^4 = 16\), and then we multiply by 5, resulting in \( f'(-2) = 80 \).This substitution process is straightforward yet powerful. It provides specific numerical results from general derivative expressions, helping to solve practical problems and gain deeper insights into how functions behave at particular points.