Problem 36 - Brief Applied Calculus

Question

31-38. Find the indicated derivatives. If \(f(x)=x^{4}\), find \(\left.\frac{d f}{d x}\right|_{x=-2}\)

Step-by-Step Solution

Answer

The derivative at \(x = -2\) is \(-32\).

Step 1: Differentiate the Function

The given function is \(f(x) = x^4\). To find the derivative, use the power rule, which states that if \(f(x) = x^n\), then \(\frac{df}{dx} = n \cdot x^{n-1}\). Applying this, we get \(f'(x) = 4 \cdot x^{4-1} = 4x^3\).

Step 2: Evaluate the Derivative at \(x = -2\)

Now, we need to evaluate the derivative we found, \(f'(x) = 4x^3\), at \(x = -2\). Substitute \(-2\) into the derivative: \(f'(-2) = 4(-2)^3 = 4 \cdot (-8) = -32\).

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus, especially crucial when learning about derivatives. It's a straightforward method for differentiating functions of the form \(f(x) = x^n\). The rule states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). This means you multiply the original exponent by the coefficient (if any) and decrease the exponent by one. This rule simplifies the process of finding derivatives for polynomial functions. For example, if you have a function \(f(x) = x^4\), using the power rule, the derivative is \(f'(x) = 4x^3\). This makes taking derivatives of polynomial terms quick and efficient. The power rule is an essential technique, especially as problems increase in complexity.

Evaluate Derivatives

Once the derivative of a function is found, it can be evaluated at a specific point to find the slope of the tangent line to the curve at that point. Evaluating a derivative at a given x-value involves substituting the value into the derived expression. For example, if we have the derivative \(f'(x) = 4x^3\) from the function \(f(x) = x^4\), evaluating it at \(x = -2\) involves substituting \(-2\) for \x\: \(f'(-2) = 4(-2)^3\). This gives us \(f'(-2) = 4 \cdot (-8) = -32\). Thus, the slope of the tangent line at \(x = -2\) is \(-32\). Evaluating derivatives is key to understanding the behavior of functions and their graphs.

Differentiation Techniques

Differentiation techniques encompass various methods to determine derivatives of functions. The power rule is one technique, but there are others such as the product rule, quotient rule, and chain rule. These techniques help differentiate more complex functions involving products, quotients, or compositions of functions. Each method is applied depending on the specific structure of the function at hand. Mastering these differentiation techniques is crucial for solving advanced calculus problems. For functions that consist of multiple types of terms or operations, choosing the correct differentiation technique is essential. Understanding when and how to apply each method aids significantly in tackling calculus problems efficiently.