/*! 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 4 Differentiate the following, sim... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 4

Differentiate the following, simplifying the expression first if useful. (a) \(y=\pi e^{3 t^{2}+\pi}\) (b) \(y=\ln \left(e^{t}+1\right)\) (c) \(y=\frac{\pi^{2}}{\sqrt{x^{2}+4}}\) (d) \(y=\frac{1}{(\ln x)^{2.6}}\) (e) \(y=\frac{1}{\left(\ln x^{2}\right)^{1.5}}\) (f) \(y=\sqrt[3]{\ln \left(e^{t}+1\right)}\)

Short Answer

Expert verified
The derivatives are (a) \(y' = 6\pi te^{3 t^{2}+\pi}\), (b) \(y'=\frac{e^t}{e^t + 1}\), (c) \(y' = -\frac{\pi^2 x}{(x^2 + 4)^{3/2}}\), (d) \(y' = -\frac{2.6}{x (\ln x)^{3.6}}\), (e) \(y'=\frac{-1.5x}{(\ln x)^{2.5}}\), and (f) \(y'=\frac{e^t}{3(\ln(e^t + 1))^{2/3}(e^t + 1)}\).

Step by step solution

01

Differentiating \(y=\pi e^{3 t^{2}+\pi}\)

Use the chain rule formula for differentiation of \(\exp(f(x))\), which is \(\exp(f(x)) \cdot f'(x)\). Here \(f(t) = 3t^2 + \pi\). So the derivative is \(y' = \pi e^{3 t^{2}+\pi} \cdot (6t)\)
02

Differentiating \(y=\ln \left(e^{t}+1\right)\)

Here, use the chain rule for differentiation of \(\ln(u)\), which is \(u'/u\). With \(u = e^t + 1\), the derivative is \(y' = \frac{e^t}{e^t + 1}\)
03

Differentiating \(y=\frac{\pi^{2}}{\sqrt{x^{2}+4}}\)

Use the quotient rule and chain rule for differentiation. The quotient rule for \(f/g\) is \((f'g - fg')/g^2\) and chain rule for \(\sqrt{u}\) is \(0.5u^{-0.5}u'\). So, \(y' = -\frac{\pi^2 x}{(x^2 + 4)^{3/2}}\)
04

Differentiating \(y=\frac{1}{(\ln x)^{2.6}}\)

Use the chain rule and the rule for differentiating the natural logarithm here. The derivative is \(y' = -\frac{2.6}{x (\ln x)^{3.6}}\)
05

Differentiating \(y=\frac{1}{\left(\ln x^{2}\right)^{1.5}}\)

Again use the chain rule and the rule for differentiating the natural logarithm. Here, \(y' = -\frac{1.5x}{(\ln x)^{2.5}}\)
06

Differentiating \(y=\sqrt[3]{\ln \left(e^{t}+1\right)}\)

Use the chain rule and the rule for differentiating the cube root and natural logarithm. Here, \(y'=\frac{e^t}{3(\ln(e^t + 1))^{2/3}(e^t + 1)}\).

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.

Chain Rule
To understand the chain rule, imagine you're peeling multiple layers to get to the core. In calculus, the chain rule is essential when you have a function nested within another function. It helps us differentiate composites of functions easily.

Consider the differentiation of the function \( y = \pi e^{3t^2 + \pi} \). Here, we have an outer exponential function and an inner function \( f(t) = 3t^2 + \pi \). The chain rule in general form is:
  • If \( y = u(v(x)) \), then \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} \).
  • For \( y = \pi e^{3t^2+\pi} \), we let \( u = f(t) = 3t^2+\pi \).
  • The derivative becomes \( y' = \pi e^{3t^2+\pi} \cdot 6t \).
The chain rule allows us to focus on differentiating the inside function first and applying the result to the outside function.
Quotient Rule
The quotient rule is indispensable when dealing with ratios of two functions. It's like applying a special formula that keeps it neat and organized. You only need to remember the rule and apply it systematically.

The formula for differentiating \( \frac{f(x)}{g(x)} \) is:
  • \( \left(\frac{f}{g}\right)' = \frac{f' \cdot g - f \cdot g'}{g^2} \)
For the function \( y = \frac{\pi^2}{\sqrt{x^2 + 4}} \), we identify:
  • \( f = \pi^2 \) and \( g = \sqrt{x^2 + 4} \).
  • The derivative of \( g \) using the chain rule is \( 0.5(x^2 + 4)^{-0.5} \cdot 2x = \frac{x}{\sqrt{x^2+4}} \).
  • Plug these into the quotient rule to get \( y' = -\frac{\pi^2 x}{(x^2 + 4)^{3/2}} \).
Exponential Functions
Exponential functions are the powerhouses of natural growth and decay. They involve constants like \( e \), the base of the natural logarithm, which is about 2.718.

An exponential function can be of the form \( y = e^{f(x)} \). Their differentiation relies heavily on the chain rule.
  • When differentiating \( y = e^{3t^2+\pi} \), recognize that \( e^{f(t)} \) keeps its form, \( e^{3t^2+\pi} \).
  • Multiply by the derivative of the exponent, resulting in \( y' = e^{3t^2+\pi} \cdot 6t \) as the chain rule dictates.
Exponential functions grow very quickly or slowly, depending on whether the exponent is positive or negative.
Logarithmic Differentiation
Logarithmic differentiation can transform complex expressions into simpler ones. It's especially useful when dealing with products, quotients, or powers raised to a variable.

Consider the function \( y = \ln(e^t + 1) \). To differentiate, use the basic property of logarithms and the chain rule:
  • The derivative formula \( \frac{d}{dx} \ln(u) = \frac{u'}{u} \) is applied.
  • For \( \ln(e^t + 1) \), \( u = e^t + 1 \) and \( u' = e^t \).
  • Thus, \( y' = \frac{e^t}{e^t + 1} \).
If you have a logarithmic function combined with others, logarithmic differentiation helps to break it down into simpler derivatives.

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

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=f\left(x^{2}\right) e^{g(x)} $$

In Problems 23 through 29, differentiate. In Problems 23 through 25, assume \(f\) is differentiable. Your answers may be in terms of \(f\) and \(f^{\prime} .\) Let \(f(x)=x^{x}\). (a) Use numerical methods to approximate \(f^{\prime}(2)\). (b) Refer to your answer to part (a) to show that \(f^{\prime}(x) \neq x \cdot x^{x-1} .\) What is it about \(f\) that makes it not a power function? (c) Refer to your answer to part (a) to show that \(f^{\prime}(x) \neq \ln x \cdot x^{x} .\) What is it about \(f\) that makes it not an exponential function? (d) Challenge: Figure out how to rewrite \(x^{x}\) so you can use the Chain Rule to differentiate it

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\frac{\pi^{2}}{3\left(x^{3}+2\right)^{6}} $$

Graph \(f(x)=e^{-x^{2}}\) and answer the following questions. (a) What is the domain of \(f ?\) The range? (b) Is \(f\) an even function, an odd function, or neither? (c) For what values of \(x\) is \(f\) increasing? Decreasing? (d) Find all relative maximum and minimum points. (e) Does \(f\) have an absolute maximum value? An absolute minimum value? A greatest lower bound? If any of these exist, identify them. (f) Find the \(x\) -coordinates of the points of inflection.

Assume that \(f, g\), and \(h\) are differentiable. Differentiate \(p(x)\) where (a) \(p(x)=f(x) g(x) h(x) .\) (Hint: Use the Product Rule twice.) (b) \(p(x)=\sqrt{g(x)+\ln f(x)}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics 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.