Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The derivative of the function is \( D_x y = 12x^3 + 3x^2 \).

Step 1: Identify the Function and its Components

The function to differentiate is given by \( y = 3x^4 + x^3 \). It consists of two terms: 1) \( 3x^4 \) and 2) \( x^3 \).

Step 2: Apply the Power Rule to Each Term

The power rule of differentiation states that \( \frac{d}{dx}(x^n) = nx^{n-1} \). We apply this rule to each term:- For \( 3x^4 \), apply the power rule: \( \frac{d}{dx}(3x^4) = 3 \times 4x^{4-1} = 12x^3 \).- For \( x^3 \), apply the power rule: \( \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 \).

Step 3: Combine the Results

Combine the derivatives of each term to find the derivative of the entire function:\( D_x y = 12x^3 + 3x^2 \).

Key Concepts

Power Rule

The Power Rule is one of the most straightforward and useful differentiation rules in calculus. It simplifies the process of finding derivatives when dealing with polynomial functions. This rule can be applied when you have a function with a variable raised to a power. Simply put, if you have a function of the form \[ f(x) = x^n \] where \(n\) is any real number, the Power Rule states that the derivative of \(f(x)\) is \[ f'(x) = nx^{n-1} \] This means you multiply the power \(n\) by the coefficient in front of \(x\), then decrease the power by one.For example, if your function is \(3x^4\), applying the power rule involves:

• Multiplying the exponent (4) by the coefficient (3) to get 12.
• Reducing the exponent by one to get \(x^3\).

That gives us the derivative, \(12x^3\). It is that simple and effective for polynomial terms.

Derivative of Polynomial Functions

A polynomial function, like \(3x^4 + x^3\), is made up of terms that are sums of variables raised to whole number powers, each with a constant coefficient. When differentiating polynomials, you differentiate term-by-term using the power rule. Each term is treated independently, and their derivatives are then summed up to get the final result.To illustrate, consider each term in the polynomial function separately:

• For the term \(3x^4\), its derivative is \(12x^3\) as calculated by applying the power rule.
• For the term \(x^3\), applying the power rule gives the derivative \(3x^2\).

Once you have the derivatives for each term, you simply add them up: \[ D_x y = 12x^3 + 3x^2 \] This is why the power rule, combined with term-by-term differentiation, makes working with polynomial functions relatively simple.

Step-by-Step Differentiation

When faced with a polynomial function, completing differentiation step-by-step ensures accuracy and understanding. Here’s how to break it down:First, identify the terms of the polynomial that need differentiation. For instance, if you have \(y = 3x^4 + x^3\), you recognize you have two distinct terms: \(3x^4\) and \(x^3\).Apply the power rule to each term. For \(3x^4\), the derivative is found by multiplying the exponent (4) by the coefficient (3), resulting in 12, and then subtracting one from the exponent. Similarly, for \(x^3\), the exponent is multiplied by its coefficient (in this case 1), giving 3, and the exponent is reduced by one.Combine the results from these applications to form the final derivative, which is the sum of the derivatives of the individual terms. Thus, the final derivative is \[ D_x y = 12x^3 + 3x^2 \] Following this sequential, logical method helps consolidate your understanding of how calculus manipulates ordinary polynomial terms for differentiation.