Problem 14

Question

Find $D_{x} y$ using the rules of this section. $$ y=3 x^{4}-2 x^{3}-5 x^{2}+\pi x+\pi^{2} $$

Step-by-Step Solution

Verified

Answer

$D_{x}y = 12x^3 - 6x^2 - 10x + \pi$.

1Step 1: Identify each term for differentiation

The function given is $y = 3x^4 - 2x^3 - 5x^2 + \pi x + \pi^2$. Each term should be treated separately for differentiation: $3x^4, -2x^3, -5x^2, \pi x, \pi^2$.

2Step 2: Differentiate each term separately

Apply the power rule for differentiation, which generally is $D_{x}(x^n) = nx^{n-1}$.1. For $3x^4$: Differentiating gives $12x^3$.2. For $-2x^3$: Differentiating gives $-6x^2$.3. For $-5x^2$: Differentiating gives $-10x^1 = -10x$.4. For $\pi x$: Differentiating gives $\pi$.5. For $\pi^2$: The derivative of a constant is 0.

3Step 3: Combine the results of differentiation

Combine the results obtained from differentiating each term separately to find the derivative $D_{x}y$:$D_{x}y = 12x^3 - 6x^2 - 10x + \pi$.

Key Concepts

Power RuleDerivative of PolynomialsCalculus Techniques

Power Rule

When differentiating a function, the power rule is one of the most fundamental and useful techniques in calculus. This rule applies when you have terms in the form of $x^n$, where $n$ is a real number. The power rule states that the derivative of $x^n$ is $nx^{n-1}$. This makes the process of finding derivatives straightforward.
For example, if you have a term like $3x^4$, you can apply the power rule by multiplying the exponent $4$ by the coefficient $3$, resulting in $12x^{3}$.
Using the power rule simplifies differentiation of polynomials, making it essential for calculus students.

Derivative of Polynomials

Polynomials are algebraic expressions that consist of terms in the form of $a_n x^n$, where $a_n$ are constants and $n$ is a nonnegative integer. Differentiating polynomials involves applying calculus rules to each term individually.
This process applies the power rule term-by-term, treating each as a separate entity.

$3x^4$ differentiates to $12x^3$
$-2x^3$ differentiates to $-6x^2$
$-5x^2$ becomes $-10x$
The term with $\pi x$ becomes $\pi$, because $\pi$ is constant
The constant $\pi^2$ has a derivative of 0.

Understanding the derivative of polynomials is crucial because it lays the foundation for more advanced topics in calculus like integration and differential equations.

Calculus Techniques

Calculus is a branch of mathematics that focuses on rates of change and accumulation. Differentiation, one of its primary tools, allows us to compute the rate of change of one quantity relative to another.
The power rule and the process of deriving polynomials are basic calculus techniques that provide a mechanism to analyze various functions in mathematics and applied sciences.
By combining these techniques:

Recognize each term in the function
Apply the appropriate rule to find the derivative
Sum the results to complete the differentiation

These steps contribute not only to the exercise of finding derivatives but also enhance our understanding of how mathematical models act and change in response to different variables.

Problem 14

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