Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=3 x^{4}-2 x^{3}-5 x^{2}+\pi x+\pi^{2} $$

Step-by-Step Solution

Answer

The derivative \( D_x y = 12x^3 - 6x^2 - 10x + \pi \).

Step 1: Identify the Rules for Derivatives

To find \(D_x y\), we will need to apply the power rule for derivatives as well as the constant rule. The power rule states that \(\frac{d}{dx}[x^n] = nx^{n-1}\), and the constant rule states that \(\frac{d}{dx}[c] = 0\) if \(c\) is a constant.

Step 2: Apply the Power Rule

Apply the power rule to each term of the function \(y = 3x^4 - 2x^3 - 5x^2 + \pi x + \pi^2\):1. Derivative of \(3x^4\) is \(3 \cdot 4x^{3} = 12x^3\).2. Derivative of \(-2x^3\) is \(-2 \cdot 3x^{2} = -6x^2\).3. Derivative of \(-5x^2\) is \(-5 \cdot 2x^{1} = -10x\).4. Derivative of \(\pi x\) is \(\pi \cdot 1x^{0} = \pi\).

Step 3: Apply the Constant Rule

For the term \(\pi^2\), since it is a constant, apply the constant rule: the derivative is \(0\).

Step 4: Combine the Derivatives

Combine the derivatives of each term to find \(D_{x} y\):\(D_{x} y = 12x^3 - 6x^2 - 10x + \pi\).

Key Concepts

Power rule for derivatives

The power rule is a fundamental tool in calculus. It's used to find derivatives of terms where the variable is raised to a power. This rule makes the process of differentiation far more straightforward, especially for polynomials. The main idea is simple: if you have a term like \(x^n\), then its derivative is \(nx^{n-1}\). The exponent is multiplied by the coefficient, and then you decrease the exponent by one.

In our problem, we applied this to each term in the polynomial. For instance, the term \(3x^4\) becomes \(12x^3\) because you multiply the 4 by the 3, and then subtract 1 from the exponent. The power rule allows you to handle each term individually, making it easy to work through even the most complex polynomials.

Constant rule in calculus

The constant rule is another basic yet crucial principle in calculus. It tells us that the derivative of any constant is zero. A constant is a number that doesn't change – like \(\pi^2\). So, when you're differentiating a function, any standalone constant just disappears.

In the exercise provided, for the term \(\pi^2\), we applied the constant rule. Hence, the derivative of \(\pi^2\) is simply 0. This rule simplifies the process of finding derivatives because it allows us to ignore terms that don't depend on the variable we're differentiating with respect to. Just remember: constants do not contribute to the derivative beyond their multiplication with variables.

Polynomial differentiation

Polynomial differentiation combines the application of both the power rule and the constant rule. When faced with a polynomial expression, like the one in our exercise, each term is tackled independently.

Here’s how we do it:

• Apply the power rule to terms with variables raised to a power, adjusting the coefficient accordingly.
• Use the constant rule to eliminate constant terms, setting their derivatives to zero.

In our problem, we had terms like \(3x^4, -2x^3, -5x^2, \pi x, \text{and } \pi^2\). Applying these rules, each term was simplified to its derivative and summed. This systematic approach enables us to efficiently differentiate any polynomial, leading to the final result: \(D_{x} y = 12x^3 - 6x^2 - 10x + \pi\). Polynomial differentiation is a key technique that underpins more advanced calculus methods.