Question

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ y=\left(x^{2}-3 x+2\right)\left(x^{3}+1\right) \quad c=2 $$

Short answer

\( f'(x) = 3x^{5} + 11x^{4} - 3x^{3} - 3x^{2} + 2x - 3 \), and \( f'(2) \) ends up to be 378.

Step-by-step solution

Identify the functions

Identify the two functions that make up the product. In this case, we have: \( f(x) = x^{2}-3x+2 \) and \( g(x) = x^{3}+1 \).

Apply the product rule

To differentiate a product of two functions, we apply the product rule which is: \( (f.g)' = f'.g + f.g' \). Here, \( f'(x) = 2x-3 \) and \( g'(x) = 3x^{2} \)

Substitute f'(x) and g'(x) into the product rule

Substitute \( f'(x) \) and \( g'(x) \) we found in step 2 into the product rule: \( (f.g)'=(2x-3)(x^{3}+1) + (x^{2}-3x+2)(3x^{2}) \). Simplify this expression to get: \( f'(x) = 2x^{4} - 3x^{2} + 3x^{5} + 2x - 3 + 9x^{4} - 3x^{3} \).

Simplify \( f'(x) \)

Arrange like terms together and simplify, yielding: \( f'(x) = 3x^{5} + 11x^{4} - 3x^{3} - 3x^{2} + 2x - 3 \).

Find \( f'(c) \)

From the problem, \( c = 2 \). Substitute \( x = 2 \) into \( f'(x) \) to yield \( f'(2) = 3(2)^{5} + 11(2)^{4} - 3(2)^{3} - 3(2)^{2} + 4 - 3 \).

Key concepts

Derivatives

In calculus, derivatives play a crucial role as they measure how a function changes at any given point. Specifically, a derivative provides the "rate of change" or the "slope" of a function's graph at a specific point. For instance, if you have a function that represents distance over time, the derivative of that function gives you the speed. When working with derivatives, especially in textbook exercises, you will often need to find the derivative of a given function. This involves using rules like the product rule, which we'll cover in more detail below. Derivatives can sometimes be daunting for beginners, but understanding the steps to calculate them makes the process much easier. Getting familiar with functions and how they behave will also help you understand their derivatives better.

Product Rule

The product rule is a fundamental rule in calculus used to differentiate functions that are products of two or more simpler functions. If you have two functions, say \( f(x) \) and \( g(x) \), and you need to find the derivative of their product \( h(x) = f(x) \cdot g(x) \), the product rule comes into play. The product rule formula is given by:

• \((f \cdot g)' = f' \cdot g + f \cdot g' \)

Here, \( f'(x) \) and \( g'(x) \) are the derivatives of \( f(x) \) and \( g(x) \) respectively. This rule ensures that you correctly account for the rates of change of both functions involved. It's essential to apply the product rule correctly in problems to find accurate derivatives of combined functions.

Differentiation

Differentiation is the process in calculus of finding the derivative of a function. It involves determining how a function's output value changes as its input value changes. Differentiating a function tells you the function's rate of change, which is valuable in many real-world applications like physics and economics.The differentiation process employs various rules, such as the product rule mentioned earlier, the chain rule, and others, depending on the structure of the function. In the original exercise, differentiation was used on the function \( y=(x^2 - 3x + 2)(x^3 + 1) \) by applying the product rule and finding the derivatives of the individual components before combining them. Understanding the steps involved in differentiation can reveal a lot about the behavior of functions based on their graphs and applications.

Functions

Functions in mathematics are relationships between a set of inputs and outputs, where each input is related to exactly one output. They are fundamental constructs in calculus and many areas of mathematics. A function can be represented in many forms, such as equations, graphs, or tables, and essential terms like domain, range, independent and dependent variables are associated with functions.In the exercise above, two functions \( f(x) = x^{2}-3x+2 \) and \( g(x) = x^{3}+1 \) were identified. Understanding their behavior separately is crucial before combining them using rules like the product rule. When dealing with functions, always look for key characteristics such as intercepts, slopes, and curves. It helps to have a solid grasp of functions when applying rules like differentiation, as this underpins most calculus operations.