Chapter 33: Problem 20
If \(f(x)=4 x^{5}-2 x^{3}+x-3\), find \(f^{\prime \prime}(x)\)
Short Answer
Expert verified
The second derivative is \(f''(x) = 80x^3 - 12x\).
Step by step solution
01
Find the First Derivative
To find the first derivative, denoted as \(f'(x)\), apply differentiation to each term in the function \(f(x) = 4x^5 - 2x^3 + x - 3\). For a term of the form \(ax^n\), the derivative is \(nax^{n-1}\). Thus, for each term:- The derivative of \(4x^5\) is \(5 \times 4x^{5-1} = 20x^4\).- The derivative of \(-2x^3\) is \(-3 \times 2x^{3-1} = -6x^2\).- The derivative of \(x\) is \(1x^{1-1} = 1\).- The derivative of a constant, \(-3\), is \(0\) as constants have no rate of change.
02
Write the First Derivative
Combine the differentiated terms to get the first derivative:\[ f'(x) = 20x^4 - 6x^2 + 1 \]
03
Find the Second Derivative
Differentiate the first derivative \(f'(x) = 20x^4 - 6x^2 + 1\) to find the second derivative, \(f''(x)\):- The derivative of \(20x^4\) is \(4 \times 20x^{4-1} = 80x^3\).- The derivative of \(-6x^2\) is \(2 \times -6x^{2-1} = -12x\).- The derivative of the constant \(1\) is \(0\).Thus, add these terms together to get \(f''(x)\).
04
Write the Second Derivative
Combine the differentiated terms to find the second derivative:\[ f''(x) = 80x^3 - 12x \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
First Derivative
The first derivative of a function helps us understand the rate of change, or how a function changes as its input changes. When you see a function like \( f(x) = 4x^5 - 2x^3 + x - 3 \), you are looking at a polynomial function comprised of multiple terms, each with its own x-power.
To find the first derivative, denoted \( f'(x) \), we perform differentiation. Differentiation is the process of finding the derivative. For polynomial terms, you use the rule that if \( ax^n \) is a term in your function, the derivative is \( nax^{n-1} \). This means you multiply the current power by the coefficient and reduce the power by 1.
To find the first derivative, denoted \( f'(x) \), we perform differentiation. Differentiation is the process of finding the derivative. For polynomial terms, you use the rule that if \( ax^n \) is a term in your function, the derivative is \( nax^{n-1} \). This means you multiply the current power by the coefficient and reduce the power by 1.
- The term \( 4x^5 \) becomes \( 20x^4 \).
- The term \( -2x^3 \) becomes \( -6x^2 \).
- The term \( x \) simply becomes 1, because the x-power of 1 is reduced to 0.
- The constant term \(-3\) does not change, as constants have no variable x to differentiate. Its rate of change is 0.
Differentiation
Differentiation is a key concept in calculus that involves finding the derivative of a function. The derivative tells us the rate at which a function's output value is changing at any given point. This is particularly useful in understanding how a polynomial function behaves as its input values change.
When differentiating a polynomial function, it’s important to remember some basic rules such as:
When differentiating a polynomial function, it’s important to remember some basic rules such as:
- Power Rule: To differentiate \( ax^n \), multiply \( a \) by \( n \) and reduce the power of \( x \) by one. For example, the derivative of \( 4x^5 \) is \( 5 \times 4x^{5-1} = 20x^4 \).
- Constants: The derivative of a constant term, such as \(-3\), is always 0 because it doesn’t change.
Polynomial Function
A polynomial function is a mathematical expression consisting of terms that are powers of \( x \), each multiplied by a coefficient. Polynomial functions are versatile and can represent a wide range of curves and shapes, making them a central object of study in algebra and calculus.
In our exercise, we have the polynomial \( f(x) = 4x^5 - 2x^3 + x - 3 \). Each term is an instance of the general form \( ax^n \), where:
In our exercise, we have the polynomial \( f(x) = 4x^5 - 2x^3 + x - 3 \). Each term is an instance of the general form \( ax^n \), where:
- \( 4x^5 \) is the leading term, with the highest power of 5.
- \( -2x^3 \) and \( x \) are middle terms with powers of 3 and 1, respectively.
- \(-3\) is the constant term that shifts the curve vertically.