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Magazine Chapter 3: Problem 9
Find \(d y / d x.\) $$y=a x^{3}+b x^{2}+c x+d \quad \text { (a. } b, c, d \text { constant) }$$
Short Answer
Expert verified
The derivative is \(3ax^2 + 2bx + c\).
Step by step solution
01
Recognize the Problem Type
The exercise requires finding the derivative of a polynomial function of the form \(y = ax^3 + bx^2 + cx + d \), where \(a, b, c,\) and \(d\) are constants. The goal is to determine \(\frac{dy}{dx}\), the derivative of the function with respect to \(x\).
02
Apply Power Rule to First Term
The first term of the polynomial is \(ax^3\). Applying the power rule of differentiation, which says \(\frac{d}{dx}[x^n] = nx^{n-1}\), we get \(\frac{d}{dx}[ax^3] = 3ax^2\).
03
Differentiate Second Term
For the second term, \(bx^2\), apply the power rule: \(\frac{d}{dx}[bx^2] = 2bx^{1} = 2bx\).
04
Differentiate Third Term
The third term is a linear term \(cx\). Using the rule \(\frac{d}{dx}[cx] = c\), we find its derivative is simply \(c\).
05
Differentiate Constant Term
The constant term \(d\) has a derivative of zero, since \(\frac{d}{dx}[d] = 0\) (the derivative of a constant is always zero).
06
Combine Derivatives
Combine all the derivatives obtained from each term: \(\frac{dy}{dx} = 3ax^2 + 2bx + c\). This is the derivative of the function \(y\) with respect to \(x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule
In calculus, the power rule is a fundamental principle used for finding the derivative of a function that is a power of a variable. This rule is essential when dealing with polynomial functions. The power rule states that if you have a function of the form \(x^n\), its derivative is given by \( \frac{d}{dx}[x^n] = nx^{n-1} \). This means you multiply the original exponent by the coefficient in front of \(x\), and then decrease the exponent by one.
For example, if you have \(x^3\), applying the power rule gives \(3x^2\). This simple rule is the building block for differentiating more complex polynomial expressions.
For example, if you have \(x^3\), applying the power rule gives \(3x^2\). This simple rule is the building block for differentiating more complex polynomial expressions.
Polynomial Differentiation
Polynomial differentiation is the process of finding the derivative of a polynomial function. Polynomials are mathematical expressions involving a sum of powers of one or more variables, such as \(ax^3 + bx^2 + cx + d\). Differentiating polynomials involves applying the power rule to each term of the polynomial.
To differentiate a polynomial:
To differentiate a polynomial:
- Identify each term individually.
- Apply the power rule to each term involving a variable.
- Combine the derivatives of each term to find the final result.
Constant Rule
The constant rule in calculus is equally straightforward and important. It states that the derivative of a constant term is always zero. This is because a constant does not change, so its rate of change is zero.
In a polynomial function, constant terms appear without a variable, like the \(d\) in the polynomial \( ax^3 + bx^2 + cx + d \). When differentiating, remember:
In a polynomial function, constant terms appear without a variable, like the \(d\) in the polynomial \( ax^3 + bx^2 + cx + d \). When differentiating, remember:
- Derivate of \(d\) is \(0\).
Calculus Basics
Calculus is the branch of mathematics that deals with the study of change. It primarily focuses on two major concepts: differentiation and integration. Differentiation, which we are focusing on here, is the process of finding the derivative of a function. A derivative represents the rate at which one quantity changes with respect to another.
Basic concepts in calculus include:
Basic concepts in calculus include:
- The derivative, which signifies change or slope.
- Rules for differentiation like the power rule.
- Simplifying expressions by recognizing constants.
