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Chapter 2: Problem 18

find the third derivative of the function. $$ f(x)=x^{4}-2 x^{3} $$

Short Answer

Expert verified
The third derivative of the function \(f(x) = x^{4} - 2x^{3}\) is \(f'''(x) = 24x - 12\).

Step by step solution

01

Find the first derivative

The original function is \(f(x) = x^{4} - 2x^{3}\). By applying the power rule for differentiation, the first derivative of the function becomes: \(f'(x) = 4x^{3} - 6x^{2}\)
02

Find the second derivative

Next, find the second derivative by taking the derivative of the first derivative. Apply the power rule again to get: \(f''(x) = 12x^{2} - 12x\)
03

Find the third derivative

Finally, take the derivative of the second derivative to get the third derivative. Once again apply the power rule to get: \(f'''(x) = 24x - 12\)

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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 a fundamental tool in calculus for finding derivatives. It simplifies the process of differentiation, especially when dealing with polynomials. To apply the power rule, if you have a term in the form of \(x^n\), the derivative is \(nx^{n-1}\). This means you multiply the entire term by the exponent and then decrease the exponent by one.
For example:
  • For \(x^4\), the derivative becomes \(4x^{3}\).
  • For \(-2x^3\), the derivative becomes \(-6x^{2}\).
The power rule is incredibly useful because it turns complex polynomial terms into easily manageable components. Remember, you apply this rule term by term across the polynomial expression.
Higher Order Derivatives
Higher order derivatives refer to taking the derivative of a function multiple times. The first derivative of a function, often noted as \(f'(x)\), gives us the rate of change or the slope of the function. If you continue to take the derivative of this first derivative, you obtain the second derivative \(f''(x)\), which can tell us things like the concavity of the graph.
The process continues in the same manner for higher orders:
  • The third derivative, \(f'''(x)\), can provide insights into the rate of change of the rate of change. This is particularly interesting in understanding the motion of physical systems, where it could describe "jerk," or how quickly acceleration is changing.
By applying the power rule multiple times, as shown in the steps with our example, these derivatives can be calculated sequentially to uncover these deeper insights into the function's behavior.
Polynomial Derivatives
When working with polynomial derivatives, we simply apply derivative rules successively to each term. Polynomials are made up of terms that are sums of variables raised to constant powers, each potentially multiplied by a constant factor.
For example, in our original function \(f(x) = x^4 - 2x^3\), finding derivatives is straightforward because each term fits the power rule format. The beauty of polynomial derivatives lies in their:
  • Predictability: Since polynomials are simple sums of these terms, the rules apply uniformly, making calculus with polynomials more manageable.
  • Applicability: Many real-world problems can be approximated or modeled using polynomial functions, making understanding their derivatives highly practical.
The step-by-step differentiation essentially breaks down a complex function into smaller, easier parts by applying familiar rules like the power rule repeatedly.

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Most popular questions from this chapter

Find the marginal revenue for producing units. (The revenue is measured in dollars.) $$ R=50\left(20 x-x^{3 / 2}\right) $$

Use Example 6 as a model to find the derivative. $$ y=\frac{\sqrt{x}}{x} $$

Find the marginal cost for producing units. (The cost is measured in dollars.) $$ C=55,000+470 x-0.25 x^{2}, \quad 0 \leq x \leq 940 $$

Marginal Profit The profit \(P(\text { in dollars) from selling } x\) units of a product is given by $$P=36,000+2048 \sqrt{x}-\frac{1}{8 x^{2}}, \quad 150 \leq x \leq 275$$ Find the marginal profit for each of the following sales. $$ \begin{array}{ll}{\text { (a) } x=150} & {\text { (b) } x=175 \quad \text { (c) } x=200} \\ {\text { (d) } x=225} & {\text { (e) } x=250 \quad \text { (f) } x=275}\end{array} $$

Inventory Management The annual inventory cost for a manufacturer is given by \(C=1,008,000 / Q+6.3 Q\) where \(Q\) is the order size when the inventory is replenished. Find the change in annual cost when \(Q\) is increased from 350 to \(351,\) and compare this with the instantaneous rate of change when \(Q=350 .\)
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