/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Derivatives Find the derivative ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 40

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of \(\sqrt{x}\). $$g(w)=\left\\{\begin{array}{ll} w+5 e^{w} & \text { if } w \leq 1 \\ 2 w^{3}+4 w+5 & \text { if } w>1 \end{array}\right.$$

Short Answer

Expert verified
The derivative of the given function \(g(w)\) is: $$ g'(w)=\left\\{\begin{array}{ll} 1 + 5 e^{w} & \text { if } w \leq 1 \\\ 6w^2 + 4 & \text { if } w>1 \end{array}\right.$$

Step by step solution

01

Find the derivative for the first case (\(w \leq 1\))

The function for this case is \(w + 5e^w\). Now apply differentiation rules: $$ \frac{d}{dw}(w) = 1$$ $$ \frac{d}{dw}(5 e^w) = 5\frac{d}{dw}(e^{w}) = 5e^w$$ Now add the derivatives: $$ \frac{d}{dw}(w + 5e^w) = 1 + 5e^w $$
02

Find the derivative for the second case (\(w > 1\))

The function for this case is \(2 w^{3}+4 w+5\). We apply differentiation rules as follows: $$ \frac{d}{dw}(2 w^{3}) = 6w^2 $$ $$ \frac{d}{dw}(4 w) = 4 $$ $$ \frac{d}{dw}(5) = 0 $$ Now add the derivatives: $$ \frac{d}{dw}(2 w^{3} + 4 w + 5) = 6w^2 + 4 $$
03

Combine the derivatives

Now that we have found the derivatives for both cases, we will combine them into a piecewise function: $$ g'(w)=\left\\{ \begin{array}{ll} 1 + 5e^w & \text { if } w \leq 1 \\\ 6w^2 + 4 & \text { if } w>1 \end{array}\right.$$ The derivative of the given function \(g(w)\) is: $$ g'(w)=\left\\{\begin{array}{ll} 1 + 5 e^{w} & \text { if } w \leq 1 \\\ 6w^2 + 4 & \text { if } w>1 \end{array}\right.$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Differentiation Rules
Differentiation is a key concept in calculus, and it involves finding the rate at which a function is changing at any given point. The rules for differentiation help simplify this process for various types of functions. Here are a few basic differentiation rules applied in our exercise to find derivatives of piecewise functions:
  • Power Rule: This rule states that if you have a function in the form of \(x^n\), then its derivative is \(nx^{n-1}\). For example, the derivative of \(2w^3\) is \(6w^2\).
  • Constant Rule: The derivative of a constant (a number on its own) is always zero. For instance, the derivative of 5 is zero.
  • Sum Rule: If you have a function that is the sum of two or more functions, the derivative of the total function is the sum of each of the individual derivatives.
Using these rules, we can differentiate piecewise functions by applying the appropriate rules to each piece.
Exponential Function Derivative
The exponential function is unique because its derivative is itself. The general form of the exponential function is \(e^x\), and its derivative is also \(e^x\). In our exercise, we had to differentiate the function part \(5e^w\). Here's how it works:
  • Basic Exponential Derivative: If you have \(e^x\), the derivative is simply \(e^x\).
  • Constant Multiple Rule: When the exponential function is multiplied by a constant, you multiply the derivative by the same constant. So, the derivative of \(5e^w\) becomes \(5e^w\).
These concepts highlight how efficiently exponential functions can be differentiated, as they retain their form through differentiation.
Polynomial Function Derivative
Polynomial functions are made up of terms which are powers of a variable, like \(w^3\) or \(w\), often with coefficients. To find the derivative of polynomial functions like \(2w^3 + 4w + 5\), we apply the power and constant rules. Here's the process:
  • Power Rule Application: For each term that is a power of the variable, apply the power rule. This makes calculating each part straightforward, such as turning \(2w^3\) into \(6w^2\).
  • Constant Term Handling: Remember, the derivative of a constant is zero, so any standalone number without a variable, like \(5\), disappears in differentiation.
  • Adding Up Derivatives: Once you've differentiated each part separately, simply add them together to get the full derivative for the polynomial function.
Differentiating polynomial functions becomes easy by applying these principles, breaking down complex expressions into simple, manageable pieces.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(f(x)=x e^{2 x}\) a. Find the values of \(x\) for which the slope of the curve \(y=f(x)\) is 0 b. Explain the meaning of your answer to part (a) in terms of the graph of \(f\)

The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots.\) c. Use the functions found in part (b) to graph the given equation. \(y^{3}=a x^{2}\) (Neile's semicubical parabola)

a. Determine an equation of the tangent line and the normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises 73-78. b. Graph the tangent and normal lines on the given graph. $$3 x^{3}+7 y^{3}=10 y; \left(x_{0}, y_{0}\right)=(1,1)$$ (Graph cant copy)

The output of an economic system \(Q,\) subject to two inputs, such as labor \(L\) and capital \(K\) is often modeled by the Cobb-Douglas production function \(Q=c L^{a} K^{b} .\) When \(a+b=1,\) the case is called constant returns to scale. Suppose \(Q=1280, a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40.\) a. Find the rate of change of capital with respect to labor, \(d K / d L.\) b. Evaluate the derivative in part (a) with \(L=8\) and \(K=64.\)

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph. $$y^{2}-3 x y=2$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.