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Chapter 3: Problem 28

\(27-30\) Find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x).\) $$f(x)=x^{5 / 2} e^{x}$$

Short Answer

Expert verified
Use the product rule to find first derivative, then calculate the second derivative directly.

Step by step solution

01

Differentiate Using the Product Rule

To find the first derivative, we'll use the product rule. The product rule states that if you have a function defined as \( f(x) = u(x)v(x) \), then the derivative \( f'(x) = u'(x)v(x) + u(x)v'(x) \). Here, we have \( u(x) = x^{5/2} \) and \( v(x) = e^x \).First, differentiate \( u(x) \): \[ u'(x) = \frac{5}{2}x^{3/2} \]Then, differentiate \( v(x) \): \[ v'(x) = e^x \]Now apply the product rule:\[ f'(x) = \left(\frac{5}{2}x^{3/2}\right)e^x + x^{5/2}e^x \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Derivatives
In calculus, the concept of a derivative is fundamental. A derivative represents the rate at which a function is changing at any given point. Imagine you are driving a car, the derivative can be thought of as your speedometer, indicating how fast your car is going at each moment. One of the primary operations in calculus, taking the derivative involves determining how a function behaves as its input changes. Mathematically, if you have a function \(f(x)\), the derivative is often noted as \(f'(x)\) or \(\frac{df}{dx}\).
  • Finding a derivative is often referred to as differentiation.
  • Derivatives can be used to find slopes, velocities, and optimization points.
Understanding derivatives is crucial for analyzing and predicting real-world phenomena.
Differentiation Techniques
Differentiation is the process of finding the derivative of a function. There are various rules and techniques used to differentiate functions, each suited to different types of expressions. Some of these include:
  • Power Rule: Useful for differentiating expressions like \(x^n\), where \(n\) is any real number. The derivative is \(nx^{n-1}\).
  • Product Rule: When differentiating a function that is a product of two functions: if \(f(x) = u(x)v(x)\), then \(f'(x) = u'(x)v(x) + u(x)v'(x)\).
  • Chain Rule: Used when dealing with composite functions, allowing differentiation of nested functions.
Each method simplifies the process and ensures that even complex functions can be analyzed effectively. By applying these rules, you can systematically work through problems without getting overwhelmed.
Connecting Derivatives with Calculus
Calculus is the overarching branch of mathematics that deals with change and motion. It encompasses both derivatives and integrals. Derivatives focus on finding the slope of a line tangent to a function's graph, representing instantaneous rates of change. Within calculus, finding derivatives is known as differential calculus. Understanding this concept is crucial, as it serves as the foundation for solving various real-world challenges:
  • Physics: calculating velocity and acceleration.
  • Economics: optimizing profits by analyzing cost functions.
  • Biology: modeling population growth over time.
The ability to differentiate allows us to dissect complex systems, making calculus an invaluable tool across multiple disciplines.
Working with Exponential Functions
Exponential functions are a key concept in calculus and are characterized by having a constant base raised to a variable exponent, typically in the form \(e^x\). These types of functions appear frequently in real-life scenarios such as population growth, radioactive decay, and compound interest calculations. Differentiating exponential functions is straightforward due to their unique property: the derivative of \(e^x\) is \(e^x\).
  • This property makes exponential functions easy to work with in calculus.
  • When combined with other functions, such as in our exercise, the product rule is essential for differentiation.
Understanding how to handle exponential functions is critical for anyone delving into calculus, as they frequently arise in both academic and practical applications.

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Most popular questions from this chapter

A flexible cable always hangs in the shape of a catenary \(y=c+a \cosh (x / a),\) where \(c\) and a are constants and a \(>0\) (see Figure 4 and Exercise 52 ). Graph several members of the family of functions \(y=a \cosh (x / a) .\) How does the graph change as a varies?

Find the derivative. Simplify where possible. $$y=x \sinh ^{-1}(x / 3)-\sqrt{9+x^{2}}$$

$$ \begin{array}{l}{\text { The radius of a circular disk is given as } 24 \mathrm{cm} \text { with a maxi- }} \\ {\text { mum error in measurement of } 0.2 \mathrm{cm} .} \\ {\text { (a) Use differentials to estimate the maximum error in the }} \\ {\text { calculated area of the disk. }} \\ {\text { (b) What is the relative error? What is the percentage error? }}\end{array} $$

A kite 100 ft above the ground moves horizontally at a speed of 8 \(\mathrm{ft} / \mathrm{s} .\) At what rate is the angle between the string and the horizontal decreasing when 200 \(\mathrm{ft}\) of string has been let out?

(a) Show that any function of the form $$\begin{array}{c}{y=A \sinh m x+B \cosh m x} \\ {\text { satisfies the differential equation } y^{\prime \prime}=m^{2} y}\end{array}$$ (b) Find \(y=y(x)\) such that \(y^{\prime \prime}=9 y, y(0)=-4\) \(\quad\) and \(y^{\prime}(0)=6\).
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