Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN: 9780321964038
Author: GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher: Pearson Addison Wesley,
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Chapter 4.6, Problem 1YT

YOUR TURN Find the derivative of y = 5 sin ⁡ 3 x 4 .

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Chapter 4 Solutions

Calculus For The Life Sciences

Ch. 4.1 - YOUR TURN If ft=1t, find f't.Ch. 4.1 - Prob. 2YTCh. 4.1 - YOUR TURN If ht=-3t2+2t+5t4-7, find h't.Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...
Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Prob. 17ECh. 4.1 - Prob. 18ECh. 4.1 - Prob. 19ECh. 4.1 - Prob. 20ECh. 4.1 - Prob. 21ECh. 4.1 - Prob. 22ECh. 4.1 - Which of the following describes the derivative...Ch. 4.1 - Which of the following describes the derivative...Ch. 4.1 - Explain the relationship between the slope and the...Ch. 4.1 - Prob. 26ECh. 4.1 - Find each derivative. Dx9x-1/2+2x3/2Ch. 4.1 - Find each derivative. Dx8x4-3x3Ch. 4.1 - Prob. 29ECh. 4.1 - Find each derivative. f'3 if fx=x39-7x2Ch. 4.1 - In Exercises 31-34, find the slope of the tangent...Ch. 4.1 - In Exercises 31-34, find the slope of the tangent...Ch. 4.1 - In Exercises 31-34, find the slope of the tangent...Ch. 4.1 - In Exercises 31-34, find the slope of the tangent...Ch. 4.1 - Prob. 35ECh. 4.1 - Find all points on the graph of fx=x3+9x2+19x-10...Ch. 4.1 - Prob. 37ECh. 4.1 - In Exercises 37-40, for each function find all...Ch. 4.1 - Prob. 39ECh. 4.1 - Prob. 40ECh. 4.1 - Prob. 41ECh. 4.1 - At what point on the graph of fx=-5x2+4x-2 is the...Ch. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.1 - Prob. 45ECh. 4.1 - Prob. 46ECh. 4.1 - Prob. 47ECh. 4.1 - In Exercises 43-46 of section 1.4, the effect of a...Ch. 4.1 - Show that, for any constant k, ddxfxk=f'xk.Ch. 4.1 - Prob. 50ECh. 4.1 - Cancer Insulation workers who were exposed to...Ch. 4.1 - Prob. 52ECh. 4.1 - Insect Mating Patterns In an experiment testing...Ch. 4.1 - Prob. 54ECh. 4.1 - Bighorn Sheep The cumulative horn volume for...Ch. 4.1 - Brain Mass The brain mass of a human fetus during...Ch. 4.1 - Prob. 57ECh. 4.1 - Body Mass Index The body mass index BMI is a...Ch. 4.1 - Prob. 59ECh. 4.1 - Prob. 60ECh. 4.1 - Prob. 61ECh. 4.1 - Prob. 62ECh. 4.1 - Prob. 63ECh. 4.1 - Dogs Human Age From the data printed in the...Ch. 4.1 - Prob. 65ECh. 4.1 - Prob. 66ECh. 4.1 - Prob. 67ECh. 4.1 - Prob. 68ECh. 4.1 - Prob. 69ECh. 4.1 - Prob. 70ECh. 4.1 - Prob. 71ECh. 4.1 - Prob. 72ECh. 4.1 - Postal Rates U.S postal rates have steadily...Ch. 4.1 - Money The total amount of money in circulation for...Ch. 4.1 - Prob. 75ECh. 4.2 - YOUR TURN Find the derivative of y=x3+74-x2.Ch. 4.2 - Prob. 2YTCh. 4.2 - YOUR TURN Find Dx5x-32x+73x+7.Ch. 4.2 - Prob. 1ECh. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Prob. 5ECh. 4.2 - Prob. 6ECh. 4.2 - Prob. 7ECh. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - Prob. 11ECh. 4.2 - Prob. 12ECh. 4.2 - Prob. 13ECh. 4.2 - Use the quotient rule to find the derivative of...Ch. 4.2 - Use the quotient rule to find the derivative of...Ch. 4.2 - Prob. 16ECh. 4.2 - Prob. 17ECh. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - Prob. 20ECh. 4.2 - Prob. 21ECh. 4.2 - Prob. 22ECh. 4.2 - Prob. 23ECh. 4.2 - Prob. 24ECh. 4.2 - Use the quotient rule to find the derivative of...Ch. 4.2 - Prob. 26ECh. 4.2 - Use the quotient rule to find the derivative of...Ch. 4.2 - Prob. 28ECh. 4.2 - Prob. 29ECh. 4.2 - Prob. 30ECh. 4.2 - Find the error in the following work....Ch. 4.2 - Prob. 32ECh. 4.2 - Find an equation of the line tangent to the graph...Ch. 4.2 - Find an equation of the line tangent to the graph...Ch. 4.2 - Prob. 35ECh. 4.2 - Prob. 36ECh. 4.2 - Prob. 37ECh. 4.2 - Prob. 38ECh. 4.2 - Prob. 39ECh. 4.2 - For each function, find the values of x in which...Ch. 4.2 - Prob. 41ECh. 4.2 - Growth Models In Exercise 52 of section 1.5, the...Ch. 4.2 - Bacteria Population Assume that the total number...Ch. 4.2 - Work/Rest Cycles Murrells formula for calculating...Ch. 4.2 - Optimal Foraging Using data collected by zoologist...Ch. 4.2 - Cell Traction Force In a matrix of cells, the cell...Ch. 4.2 - Prob. 47ECh. 4.2 - Cellular Chemistry The release of a chemical...Ch. 4.2 - Prob. 49ECh. 4.2 - Memory Retention Some psychologists content that...Ch. 4.2 - Prob. 51ECh. 4.2 - Prob. 52ECh. 4.3 - YOUR TURN For the functions in Example 1, find fg0...Ch. 4.3 - YOUR TURN Let fx=2x-3 and gx=x2+1. Find gfx.Ch. 4.3 - YOUR TURN Write hx=2x-33 as a composition of two...Ch. 4.3 - YOUR TURN Find dy/dx if y=5x2-6x-2.Ch. 4.3 - Prob. 5YTCh. 4.3 - Prob. 6YTCh. 4.3 - Prob. 7YTCh. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.3 - Prob. 10ECh. 4.3 - In Exercises 7-14, find f[gx] and g[fx]. fx=x+2;...Ch. 4.3 - In Exercises 7-14, find f[gx] and g[fx]....Ch. 4.3 - Prob. 13ECh. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.3 - Prob. 16ECh. 4.3 - Prob. 17ECh. 4.3 - Prob. 18ECh. 4.3 - Prob. 19ECh. 4.3 - Write each function as the composition of two...Ch. 4.3 - Prob. 21ECh. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Prob. 23ECh. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Prob. 28ECh. 4.3 - Prob. 29ECh. 4.3 - Prob. 30ECh. 4.3 - Prob. 31ECh. 4.3 - Prob. 32ECh. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Prob. 35ECh. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Find the derivative of each function defined as...Ch. 4.3 - Prob. 41ECh. 4.3 - Prob. 42ECh. 4.3 - Prob. 43ECh. 4.3 - Consider the following table of values of the...Ch. 4.3 - Prob. 45ECh. 4.3 - In Exercises 45-48, find the equation of the...Ch. 4.3 - In Exercises 45-48, find the equation of the...Ch. 4.3 - In Exercises 45-48, find the equation of the...Ch. 4.3 - Prob. 49ECh. 4.3 - Prob. 50ECh. 4.3 - Katie and Sarah are working on taking the...Ch. 4.3 - Margy and Nate are working on taking the...Ch. 4.3 - Prob. 53ECh. 4.3 - Oil Pollution An oil well off the Gulf Coast is...Ch. 4.3 - African Wild Dog The African wild dog is currently...Ch. 4.3 - Thermal Inversion When there is a thermal...Ch. 4.3 - Prob. 57ECh. 4.3 - Calcium Usage To test an individuals use of...Ch. 4.3 - Drug Reaction The strength of a persons reaction...Ch. 4.3 - Extinction The probability of a population going...Ch. 4.3 - Candy The volume and surface area of a jawbreaker...Ch. 4.3 - Prob. 63ECh. 4.3 - Prob. 64ECh. 4.3 - Prob. 65ECh. 4.3 - Prob. 66ECh. 4.4 - YOUR TURN 1 Find dy/dx for a. y=43x, b. y=e7x3+5Ch. 4.4 - Prob. 2YTCh. 4.4 - Prob. 3YTCh. 4.4 - YOUR TURN 4 The quantity in grams of a radioactive...Ch. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 10ECh. 4.4 - Prob. 11ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 13ECh. 4.4 - Prob. 14ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 16ECh. 4.4 - Prob. 17ECh. 4.4 - Prob. 18ECh. 4.4 - Prob. 19ECh. 4.4 - Prob. 20ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 25ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 31ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 33ECh. 4.4 - Find derivatives of the functions defined as...Ch. 4.4 - Prob. 35ECh. 4.4 - Prob. 36ECh. 4.4 - Use graphical differentiation to verify that...Ch. 4.4 - Prob. 38ECh. 4.4 - Population Growth In section 2.1, Exercise 37, the...Ch. 4.4 - Minority Population In Section 2.1, Exercise 39,...Ch. 4.4 - Insect Growth The growth of a population of rare...Ch. 4.4 - Prob. 43ECh. 4.4 - Whooping Cranes Based on data from the U.S. Fish...Ch. 4.4 - Prob. 45ECh. 4.4 - Prob. 46ECh. 4.4 - Mortality The percentage of people of any...Ch. 4.4 - Breast Cancer It has been observed that the...Ch. 4.4 - Arctic Foxes The age-weight relationship of female...Ch. 4.4 - Prob. 50ECh. 4.4 - Ayrshire Dairy Cattle The researchers from the...Ch. 4.4 - Cutlassfish The cutlassfish is one of the most...Ch. 4.4 - Prob. 53ECh. 4.4 - Dialysis One measure of whether a dialysis patient...Ch. 4.4 - Medical Literature It has been observed that there...Ch. 4.4 - Prob. 56ECh. 4.4 - Prob. 57ECh. 4.4 - Prob. 58ECh. 4.4 - Prob. 59ECh. 4.4 - Prob. 60ECh. 4.4 - Radioactive Iron Researchers found that the...Ch. 4.4 - Prob. 62ECh. 4.4 - Prob. 63ECh. 4.4 - Prob. 64ECh. 4.4 - Habit Strength According to work by the...Ch. 4.4 - Prob. 66ECh. 4.4 - Electricity In a series resistance-capacitance DC...Ch. 4.4 - Prob. 68ECh. 4.4 - Prob. 69ECh. 4.5 - YOUR TURN Find the derivative of fx=log3x.Ch. 4.5 - Prob. 2YTCh. 4.5 - Prob. 3YTCh. 4.5 - Find the derivative of each function. y=ln8xCh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Find the derivative of each function. y=ln4x2-9xCh. 4.5 - Find the derivative of each function. y=ln-8x3+2xCh. 4.5 - Find the derivative of each function. y=lnx+5Ch. 4.5 - Prob. 8ECh. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Prob. 13ECh. 4.5 - Find the derivative of each function. y=xln2-x2Ch. 4.5 - Prob. 15ECh. 4.5 - Find the derivative of each function. v=lnuu3Ch. 4.5 - Prob. 17ECh. 4.5 - Prob. 18ECh. 4.5 - Find the derivative of each function. y=3x2lnxCh. 4.5 - Prob. 20ECh. 4.5 - Prob. 21ECh. 4.5 - Find the derivative of each function. y=lnx-3Ch. 4.5 - Find the derivative of each function. y=lnlnxCh. 4.5 - Prob. 24ECh. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - Prob. 27ECh. 4.5 - Find the derivative of each function. py=lnyeyCh. 4.5 - Prob. 29ECh. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - Find the derivative of each function. y=log4x-3Ch. 4.5 - Prob. 33ECh. 4.5 - Prob. 34ECh. 4.5 - Prob. 35ECh. 4.5 - Find the derivative of each function. y=log74x-3Ch. 4.5 - Find the derivative of each function....Ch. 4.5 - Find the derivative of each function....Ch. 4.5 - Prob. 39ECh. 4.5 - Find the derivative of each function. z=10ylogyCh. 4.5 - Prob. 41ECh. 4.5 - Find the derivative of each function. fx=lnxex+2Ch. 4.5 - Prob. 43ECh. 4.5 - Prob. 44ECh. 4.5 - Prob. 45ECh. 4.5 - Prob. 46ECh. 4.5 - Prob. 47ECh. 4.5 - Use a graphing calculator to sketch the graph of...Ch. 4.5 - Prob. 49ECh. 4.5 - Prob. 50ECh. 4.5 - Prob. 51ECh. 4.5 - Prob. 52ECh. 4.5 - Prob. 53ECh. 4.5 - Prob. 54ECh. 4.5 - Prob. 55ECh. 4.5 - This exercise shows another way to derive the...Ch. 4.5 - Prob. 57ECh. 4.5 - Population Growth Suppose that the population of a...Ch. 4.5 - Body Surface Area There is a mathematical...Ch. 4.5 - Prob. 60ECh. 4.5 - Prob. 61ECh. 4.5 - Fruit Flies A study of the relation between the...Ch. 4.5 - Prob. 63ECh. 4.5 - Poverty The passage of the Social Security...Ch. 4.5 - Richter Scale The Richter scale provides a measure...Ch. 4.5 - Prob. 66ECh. 4.6 - YOUR TURN Find the derivative of y=5sin3x4.Ch. 4.6 - YOUR TURN Find the derivative of y=2sin3x.Ch. 4.6 - Prob. 3YTCh. 4.6 - Prob. 4YTCh. 4.6 - Prob. 5YTCh. 4.6 - Prob. 6YTCh. 4.6 - Prob. 1ECh. 4.6 - Prob. 2ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 4ECh. 4.6 - Prob. 5ECh. 4.6 - Prob. 6ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 8ECh. 4.6 - Prob. 9ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 11ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 13ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 15ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 17ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 19ECh. 4.6 - Prob. 20ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 23ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 25ECh. 4.6 - Find the derivatives of the functions defined as...Ch. 4.6 - Prob. 27ECh. 4.6 - Prob. 28ECh. 4.6 - Prob. 29ECh. 4.6 - In Exercises 27-32, recall that the slope of the...Ch. 4.6 - Prob. 31ECh. 4.6 - In Exercises 27-32, recall that the slope of the...Ch. 4.6 - Find the derivative of cotx by using the quotient...Ch. 4.6 - Prob. 34ECh. 4.6 - Verify that the derivative of cscx is -cscxcotx....Ch. 4.6 - Prob. 36ECh. 4.6 - Prob. 37ECh. 4.6 - Carbon Dioxide Levels At Barrow, Alaska,...Ch. 4.6 - Prob. 39ECh. 4.6 - Sound If a string with a fundamental frequency of...Ch. 4.6 - Prob. 41ECh. 4.6 - Revenue from Seasonal Merchandise The revenue...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Prob. 10CRCh. 4.CR - Determine whether each of the following statements...Ch. 4.CR - Prob. 12CRCh. 4.CR - Prob. 13CRCh. 4.CR - Prob. 14CRCh. 4.CR - Prob. 15CRCh. 4.CR - Prob. 16CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 19CRCh. 4.CR - Prob. 20CRCh. 4.CR - Prob. 21CRCh. 4.CR - Prob. 22CRCh. 4.CR - Prob. 23CRCh. 4.CR - Prob. 24CRCh. 4.CR - Prob. 25CRCh. 4.CR - Prob. 26CRCh. 4.CR - Prob. 27CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 29CRCh. 4.CR - Prob. 30CRCh. 4.CR - Prob. 31CRCh. 4.CR - Prob. 32CRCh. 4.CR - Prob. 33CRCh. 4.CR - Prob. 34CRCh. 4.CR - Prob. 35CRCh. 4.CR - Prob. 36CRCh. 4.CR - Prob. 37CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 39CRCh. 4.CR - Prob. 40CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 42CRCh. 4.CR - Prob. 43CRCh. 4.CR - Prob. 44CRCh. 4.CR - Prob. 45CRCh. 4.CR - Prob. 46CRCh. 4.CR - Prob. 47CRCh. 4.CR - Prob. 48CRCh. 4.CR - Prob. 49CRCh. 4.CR - Prob. 50CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 52CRCh. 4.CR - Prob. 53CRCh. 4.CR - Prob. 54CRCh. 4.CR - Prob. 55CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 57CRCh. 4.CR - Prob. 58CRCh. 4.CR - Prob. 59CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 61CRCh. 4.CR - Prob. 62CRCh. 4.CR - Prob. 63CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 65CRCh. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Use the rules for derivatives to find the...Ch. 4.CR - Prob. 68CRCh. 4.CR - Consider the following table of values of the...Ch. 4.CR - Consider the following table of values of the...Ch. 4.CR - Prob. 71CRCh. 4.CR - Prob. 72CRCh. 4.CR - Find the slope of the tangent line to the given...Ch. 4.CR - Prob. 74CRCh. 4.CR - Prob. 75CRCh. 4.CR - Prob. 76CRCh. 4.CR - Prob. 77CRCh. 4.CR - Prob. 78CRCh. 4.CR - Prob. 79CRCh. 4.CR - Prob. 80CRCh. 4.CR - Prob. 81CRCh. 4.CR - Prob. 82CRCh. 4.CR - Prob. 83CRCh. 4.CR - Prob. 84CRCh. 4.CR - Prob. 85CRCh. 4.CR - Prob. 86CRCh. 4.CR - Prob. 87CRCh. 4.CR - Prob. 88CRCh. 4.CR - Prob. 89CRCh. 4.CR - Exponential Growth Suppose a population is growing...Ch. 4.CR - Logistic Growth Suppose a population can be...Ch. 4.CR - Fish The length of the monkeyface prickleback, a...Ch. 4.CR - Prob. 93CRCh. 4.CR - Prob. 94CRCh. 4.CR - Prob. 95CRCh. 4.CR - Prob. 96CRCh. 4.CR - Prob. 97CRCh. 4.CR - Prob. 98CRCh. 4.CR - Prob. 99CRCh. 4.CR - Prob. 100CRCh. 4.CR - Prob. 101CRCh. 4.EA - Suppose that a particular plot of land can sustain...Ch. 4.EA - Prob. 2EA
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Chain Rule dy:dx = dy:du*du:dx; Author: Robert Cappetta;https://www.youtube.com/watch?v=IUYniALwbHs;License: Standard YouTube License, CC-BY
CHAIN RULE Part 1; Author: Btech Maths Hub;https://www.youtube.com/watch?v=TIAw6AJ_5Po;License: Standard YouTube License, CC-BY