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Chapter 3: Problem 35

In Exercises \(33-36,\) find the first four derivatives of the function. $$y=x^{-1}+x^{2}$$

Short Answer

Expert verified
The first four derivatives of the function are: \(y^{'}=-x^{-2}+2x\), \(y^{''}=2x^{-3}+2\), \(y^{'''}=-6x^{-4}\), and \(y^{''''}=24x^{-5}\).

Step by step solution

01

Identify the Function

The function given is \(y=x^{-1}+x^{2}\).
02

Compute the First Derivative

Using the power rule, we find the derivative of \(x^{-1}\) to be \(-x^{-2}\), and of \(x^{2}\) to be \(2x\). Thus, the first derivative is \(y^{'}=-x^{-2}+2x\).
03

Compute the Second Derivative

Applying the power rule again, the derivative of \(-x^{-2}\) is \(2x^{-3}\) and of \(2x\) is \(2\). So, the second derivative is \(y^{''}=2x^{-3}+2\).
04

Compute the Third Derivative

Differentiating again, find the derivative of \(2x^{-3}\) to be \(-6x^{-4}\) and the derivative of \(2\) is \(0\). The third derivative is \(y^{'''}=-6x^{-4}\).
05

Compute the Fourth Derivative

Once more, the derivative of \(-6x^{-4}\) is \(24x^{-5}\). Therefore, the fourth derivative is \(y^{''''}=24x^{-5}\).

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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 rule in calculus, especially important for finding derivatives. It provides a quick and simple way to differentiate functions of the form \(x^n\). To apply the power rule, take the exponent \(n\), multiply it by the coefficient of \(x\), then decrease the exponent by one.
For example, to differentiate \(x^{2}\):
  • Multiply the exponent (2) by the coefficient (1), resulting in \(2\).
  • Lower the exponent by one, giving \(2-1 = 1\). Thus, the derivative is \(2x^1\) or \(2x\).
This rule applies to negative exponents as well, allowing for the differentiation of terms like \(x^{-1}\). By following the same steps, you can handle both simple monomial functions and components of more complex functions with ease.
First Derivative
Finding the first derivative of a function is the initial step in understanding its rate of change. It tells you how the function behaves as \(x\) changes. Using the power rule, we found the first derivative of the given function \(y = x^{-1} + x^{2}\) to be \(y^{'} = -x^{-2} + 2x\).

Here's how it works:
  • The derivative of \(x^{-1}\) is found by applying the power rule: multiply \(-1\) by \(x\)'s coefficient (1) to get \(-1x^{-2}\).
  • For \(x^2\), raise the constant coefficient by multiplying by the exponent, resulting in \(2x\).
The result, \(-x^{-2} + 2x\), represents the slope of the tangent line at any point \(x\) on the original function. This helps in sketching curves and analyzing intervals where the function is increasing or decreasing.
Higher-Order Derivatives
Higher-order derivatives provide deeper insight into a function's behavior through its second, third, fourth derivatives, and beyond. Each derivative gives a step into understanding more about the function like velocity, acceleration, or change of acceleration.

Once we've found the first derivative, we can find successive derivatives for further analysis:
  • **Second Derivative:** Apply the power rule to each term of the first derivative \(y' = -x^{-2} + 2x\), resulting in \(y^{''} = 2x^{-3} + 2\).
  • **Third Derivative:** Continue differentiating \(y^{''}\) to obtain \(y^{'''} = -6x^{-4}\).
  • **Fourth Derivative:** The next level of differentiation gives \(y^{''''} = 24x^{-5}\).
Each successive derivative provides insights like concavity and points of inflection that reveal more about the function's graphical representation.
Calculus Problems
Calculus problems often require derivatives to solve various types of questions, from finding maxima and minima to solving differential equations. They are integral to fields such as physics, economics, and engineering, where rates of change play a crucial role.

When tackling calculus problems, consider:
  • Using derivatives to find critical points in optimization problems. These points help determine the maximum efficiency or cost minimization, for example.
  • Applying derivatives to compute instantaneous rate of change, which is essential in motion and physics problems.
  • Solving real-world problems through predictive modeling and analysis of changing systems.
Mastering the concepts of derivatives, including the power rule and higher-order derivatives, equips you with powerful tools to tackle these calculus challenges effectively.

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