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Magazine Chapter 2: Problem 22
Find the derivative of each function. $$f(x)=\frac{1}{24} x^{4}+\frac{1}{6} x^{3}+\frac{1}{2} x^{2}+x+1$$
Short Answer
Expert verified
The derivative is \(\frac{1}{6}x^3 + \frac{1}{2}x^2 + x + 1\).
Step by step solution
01
Identify the Derivative Rules
To find the derivative of a polynomial function, you can use the power rule. The power rule states that the derivative of a term like \(a \cdot x^n\) is \(a \cdot n \cdot x^{n-1}\). We'll apply this rule to each term in the polynomial.
02
Differentiate the First Term
The first term is \(\frac{1}{24}x^4\). Apply the power rule: multiply the coefficient by the exponent (4), then subtract 1 from the exponent. This results in \(\frac{1}{24} \cdot 4 \cdot x^{3} = \frac{1}{6}x^{3}\).
03
Differentiate the Second Term
The second term is \(\frac{1}{6}x^3\). Again apply the power rule: \(\frac{1}{6} \cdot 3 \cdot x^{2} = \frac{1}{2}x^{2}\).
04
Differentiate the Third Term
The third term is \(\frac{1}{2}x^2\). Applying the power rule, we get \(\frac{1}{2} \cdot 2 \cdot x^{1} = x\).
05
Differentiate the Fourth Term
The fourth term is \(x\), which can be written as \(1x^1\). Applying the power rule, \(1 \cdot 1 \cdot x^{0} = 1\).
06
Differentiate the Constant Term
The last term is a constant, \(1\). The derivative of a constant is \(0\).
07
Combine All the Derivatives
Combine the derivatives of all the terms calculated: \(\frac{1}{6}x^3 + \frac{1}{2}x^2 + x + 1 + 0\). Simplify it to get the final answer: \(\frac{1}{6}x^3 + \frac{1}{2}x^2 + x + 1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule
The power rule is one of the fundamental rules in calculus used to find the derivative of a term that contains a variable raised to an exponent. It is a straightforward yet powerful tool for differentiation of polynomial functions. According to the power rule, if you have a term like \( a \cdot x^n \), the derivative is calculated as \( a \cdot n \cdot x^{n-1} \). This means you should multiply the constant coefficient \( a \) by the exponent \( n \), and then decrease the exponent by one.
- For instance, with the term \( 3x^4 \), apply the power rule: \( 3 \cdot 4 \cdot x^{3} = 12x^{3} \).
- Remember that this rule also applies when the exponent is 1, where you end up with the constant coefficient as the derivative since \( x^1 \) becomes \( x^0 = 1 \).
- The power rule simplifies the process of differentiation, allowing you to handle each term independently.
Polynomial Functions
Polynomial functions are expressions made up of terms in the form \( a \cdot x^n \). Each term of a polynomial is called a monomial, and the polynomial is simply the sum of these monomials. When finding derivatives, it's crucial to apply the power rule to each term independently.
- Polynomials can consist of constants, linear terms, quadratic terms, and higher powers of \( x \).
- For example, the function given in the exercise \( f(x)=\frac{1}{24} x^{4}+\frac{1}{6} x^{3}+\frac{1}{2} x^{2}+x+1 \) is a polynomial because it includes several terms with \( x \) raised to different powers.
- Each term of this polynomial can be differentiated using the power rule to find its individual derivative.
Differentiation
Differentiation is a mathematical process used to find the rate at which a function is changing at any given point. In simpler terms, it allows you to find the slope of the function's graph at a particular point. This process is popularly utilized in calculus to determine the derivative of a function.
- The derivative provides valuable insights into a function鈥檚 behavior, such as identifying local maxima and minima and determining concavity.
- To differentiate a polynomial function like \( f(x)=\frac{1}{24} x^{4}+\frac{1}{6} x^{3}+\frac{1}{2} x^{2}+x+1 \), you apply differentiation rules, particularly the power rule, to each term to find its rate of change.
- Once differentiated, the equation \( f'(x) \) gives you the mathematical expression representing the rate of change across the function \( f(x) \).
