Question

Find \(D_{x} y\) using the rules of this section. $$ y=\left(x^{2}+17\right)\left(x^{3}-3 x+1\right) $$

Short answer

The derivative is \( D_x y = 5x^4 + 42x^2 + 2x - 51 \).

Step-by-step solution

Identify the Function

The function given is a product of two expressions: \( y = (x^2 + 17)(x^3 - 3x + 1) \). We need to find the derivative of this product.

Apply the Product Rule

To differentiate the product of two functions, \( u(x) = x^2 + 17 \) and \( v(x) = x^3 - 3x + 1 \), we use the product rule: \( D_x(uv) = u'v + uv' \).

Differentiate \(u(x)\)

Find the derivative \(u'(x)\) where \( u(x) = x^2 + 17 \). The derivative \( u'(x) = 2x \), since the derivative of \( x^2 \) is \( 2x \) and the derivative of a constant is 0.

Differentiate \(v(x)\)

Find the derivative \(v'(x)\) where \( v(x) = x^3 - 3x + 1 \). The derivative \( v'(x) = 3x^2 - 3 \), because the derivative of \( x^3 \) is \( 3x^2 \) and the derivative of \( -3x \) is \( -3 \).

Substitute Derivatives into Product Rule

Using the derivatives \( u'(x) = 2x \) and \( v'(x) = 3x^2 - 3 \), substitute these into the product rule formula: \( 2x(x^3 - 3x + 1) + (x^2 + 17)(3x^2 - 3) \).

Simplify the Expression

First multiply the terms: \( 2x \times (x^3 - 3x + 1) \) results in \( 2x^4 - 6x^2 + 2x \). Then \( (x^2 + 17) \times (3x^2 - 3) \) results in \( 3x^4 + 51x^2 - 3x^2 - 51 \), which simplifies to \( 3x^4 + 48x^2 - 51 \).

Combine Like Terms

Add \( 2x^4 - 6x^2 + 2x \) and \( 3x^4 + 48x^2 - 51 \): \[ D_x y = (2x^4 + 3x^4) + (-6x^2 + 48x^2) + (2x) + (-51) \]Simplifying, \( D_x y = 5x^4 + 42x^2 + 2x - 51 \).

Key concepts

Derivatives

Derivatives are fundamental to calculus, representing the rate of change of a function with respect to a variable. When we seek the derivative of a function, we're looking for how that function changes as its input changes. This can be visualized as the slope of the function at any point.

In our original exercise, derivatives are used to find the rate of change of the product of two polynomial functions. The ability to differentiate these functions helps us understand their behavior and predict patterns.

• Instantaneous Rate of Change: The derivative gives the instant rate at which the function value is changing relative to the change in the input.
• Slope of Tangent Lines: At any given point, the value of the derivative is also the slope of the tangent line to the function. This gives us geometric insights into the function's graph.

Understanding derivatives and their calculations provide crucial tools in problem-solving across physics, engineering, economics, and various scientific fields.

Differentiation

Differentiation is the process of finding the derivative of a function. This involves applying specific rules that systematically transform a function into its derivative form. One of the critical rules we often use is the product rule, which applies when differentiating the product of two functions.

In the provided exercise, differentiation was accomplished using this product rule. The product rule states: If you have two functions, say \( u(x) \) and \( v(x) \), then the derivative of their product, \( uv \), is given by: \( D_x(uv) = u'v + uv' \).

• Chain Rule: Another rule often used alongside the product rule is the chain rule, which comes in handy when dealing with composite functions.
• Sum and Difference Rules: These rules are the easiest, allowing derivatives of sums and differences of functions to be found by differentiating each component separately.

These differentiation rules are essential in breaking down complex problems into simpler parts, making it possible to handle complicated functions.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients, involving only non-negative integer powers of the variables. They are structured in terms of ascending powers from lower to higher degrees, making them predictable and easier to manipulate.

In the exercise discussed, both expressions being differentiated, \( x^2 + 17 \) and \( x^3 - 3x + 1 \), are polynomial functions. Differentiating these involves straightforward application of power rules, ensuring each term is tackled efficiently.

• Simplified Derivatives: The derivative of \( x^n \) is \( nx^{n-1} \). This simple rule makes working with polynomials particularly handy.
• Smooth and Continuous: Polynomial functions are inherently smooth and continuous, making them preferable models for various real-world scenarios.
• Easy to Combine: The nature of polynomials allows for easy combination and simplification, such as combining like terms after differentiation.

Grasping the behaviors and properties of polynomial functions is essential for efficiently applying calculus techniques in various mathematical scenarios.