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Chapter 3: Problem 14

\(2-22\) Differentiate the function. \(F(y)=y \ln \left(1+e^{y}\right)\)

Short Answer

Expert verified
The derivative of the function is \( F'(y) = \ln(1 + e^y) + \frac{y e^y}{1 + e^y} \).

Step by step solution

01

Rewrite the Function

Consider the function given: \[ F(y) = y \ln(1+e^y) \]. This is a product of two functions, \( y \) and \( \ln(1 + e^y) \).
02

Apply the Product Rule

To differentiate \( F(y) \), use the product rule which states: \( (uv)' = u'v + uv' \). Here, let \( u = y \) and \( v = \ln(1 + e^y) \), then differentiate each function individually.
03

Differentiate \( u = y \)

The derivative of \( u = y \) with respect to \( y \) is \( u' = 1 \).
04

Differentiate \( v = \ln(1 + e^y) \)

Use the chain rule to differentiate \( v \). Let \( g(y) = 1 + e^y \), then \( v = \ln(g(y)) \). The derivative is \( v' = \frac{1}{g(y)}g'(y) = \frac{e^y}{1+e^y} \).
05

Substitute Back in the Product Rule

The product of \( u' \) and \( v \) is \( 1 \times \ln(1 + e^y) \). The product of \( u \) and \( v' \) is \( y \times \frac{e^y}{1 + e^y} \).
06

Combine Results for Final Derivative

The derivative of \( F(y) \) is given by combining the results from Step 5:\[ F'(y) = \ln(1 + e^y) + \frac{y e^y}{1 + e^y} \].

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Key Concepts

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

Product Rule
The product rule is essential when differentiating a function composed of two multiplying parts. It's a powerful tool in calculus differentiating functions of the form \( u(y) \cdot v(y) \). When tackling these problems, think of the product rule as needing two derivative evaluations to find the answer. Consider functions \( u \) and \( v \). The product rule formula is:
  • Derivative of the product: \( (uv)' = u'v + uv' \)
  • First, differentiate \( u \) separately: \( u' \)
  • Then, differentiate \( v \) separately: \( v' \)
  • Finally, substitute back into the formula: add \( u'v \) to \( uv' \)
Applying this to the exercise, \( y \) is \( u \) and \( \ln(1+ e^y) \) is \( v \). Differentiating each individually and inserting them into the formula gives the final derivative of the function. This breaks the process into simple, manageable steps, ensuring the precise result.
Chain Rule
The chain rule is another fundamental principle for differentiating composite functions. When a function is inside another function, like \( \ln(1+e^y) \), the chain rule becomes your go-to technique. Consider the chain rule as taking the derivative step by step, starting with the outer function and working inwards. The formula is:
  • Derivative of a composition: \( (f \circ g)'(x) = f'(g(x)) \cdot g'(x) \)
  • Start by differentiating the outer function \( f \)
  • Multiply by the derivative of the inner function \( g \)
In the given solution, \( \ln(g(y)) \) is more accessible when substituting \( g(y) = 1+e^y \). You first take the derivative of \( \ln(x) \) as \( \frac{1}{x} \), then multiply by the derivative of the inner function, which results in \( \frac{e^y}{1+e^y} \). This simplifies the process and guarantees accurately finding the derivative.
Derivatives
Derivatives provide a mathematical way of expressing how a function changes as its input changes. They are central to calculus and represent many real-world phenomena such as rates of change and slopes of curves. Think of derivatives as a snapshot of the way a function behaves at any given point.
  • The basic derivative rules include constants becoming zero, \( x^n \) becoming \( nx^{n-1} \), and constant multiples
  • Real-life applications could be the velocity of an object, which is the derivative of its position over time
Derivatives like \( y \)'s to \( 1 \) are straightforward, providing a solid grounding to work from. In our problem, rapid computation occurs for simple terms in the derivative formula, concentrating more on complex aspects like methods using rules like product and chain. Mastery of derivatives lets you handle complex calculus operations smoothly.

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Most popular questions from this chapter

Find the derivative. Simplify where possible. $$f(x)=x \sinh x-\cosh x$$

$$ \begin{array}{c}{\text { When blood flows along a blood vessel, the flux F (the }} \\ {\text { volume of blood per unit time that flows past a given point) }} \\ {\text { is proportional to the fourth power of the radius R of the }} \\ {\text { blood vessel: }} \\ {\quad F=k R^{4}}\end{array} $$ This is known as Poiseuille's Law; we will show why it is true in Section 8.4 . ) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in \(F\) is about four times the relative change in R. How will a 5\(\%\) increase in the radius affect the flow of blood?

If \(R\) denotes the reaction of the body to some stimulus of strength \(x,\) the sensitivity \(S\) is defined to be the rate of change of the reaction with respect to \(x\) . A particular example is that when the brightness \(x\) of a light source is increased, the eye reacts by decreasing the area \(R\) of the pupil. The experimental formula $$R=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$$ has been used to model the dependence of \(\mathrm{R}\) on \(\mathrm{x}\) when \(\mathrm{R}\) is measured in square millimeters and \(\mathrm{x}\) is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both \(\mathrm{R}\) at low levels of of \(\mathrm{x} .\) Comment on the values of \(\mathrm{R}\) and \(\mathrm{S}\) at low levels of brightness. Is this what you would expect?

Brain weight \(\mathrm{B}\) as a function of body weight \(\mathrm{W}\) in fish has been modeled by the power function \(\mathrm{B}=0.007 \mathrm{W}^{2 / 3}\) , where \(\mathrm{B}\) and \(\mathrm{W}\) are measured in grams. A model for body weight as a function of body length \(\mathrm{L}\) (measured in centimeters) is \(\mathrm{W}=0.12 \mathrm{L}^{2.53}\) . If, over 10 million years, the average length of a certain species of fish evolved from 15 \(\mathrm{cm}\) to 20 \(\mathrm{cm}\) at a constant rate, how fast was this species' brain growing when the average length was 18 \(\mathrm{cm} ?\)

A particle is moving along the curve \(y=\sqrt{x}\) . As the particle passes through the point \((4,2),\) its x-coordinate increases at a rate of 3 \(\mathrm{cm} / \mathrm{s}\) . How fast is the distance from the particle to the origin changing at this instant?
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