Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 3.5, Problem 1E

$(a)$

To determine

To calculate: The first derivative of the function $xy+2x+3x2=4$ by implicit differentiation.

$(a)$

Expert Solution

Answer to Problem 1E

The first derivative of the function is $y'=−y−6x−2x$ .

Explanation of Solution

Given information:

The function $xy+2x+3x2=4$ .

Formula used:

Thechain rule for differentiation is if f is a function of gthen $ddx(f(g(x)))=f'(g(x))g'(x)$ .

Power rule for differentiation is $ddxxn=nxn−1$ .

Product rule for differentiation is $ddx(f⋅g)=f'(x)g(x)+f(x)g'(x)$ where f and g are functions of x .

Calculation:

Consider the function $xy+2x+3x2=4$ .

Differentiate both sides with respect to x ,

$ddx(xy+2x+3x2)=ddx(4)ddx(xy)+ddx(2x)+ddx(3x2)=0$

Recall that power rule for differentiation is $ddxxn=nxn−1$ and chain rule for differentiation is if f is a function of gthen $ddx(f(g(x)))=f'(g(x))g'(x)$ .

Apply it. Also observe that y is a function of x,

$ddx(xy+2x+3x2)=ddx(4)ddx(xy)+ddx(2x)+ddx(3x2)=0y+xy'+2+6x=0$

Isolate the value of $y'$ on left hand side and simplify,

$y+xy'+2+6x=0xy'=−y−6x−2y'=−y−6x−2x$

Thus, the first derivative of the function is $y'=−y−6x−2x$ .

$(b)$

To determine

To calculate: The first derivative of the function $xy+2x+3x2=4$ by explicit differentiation.

$(b)$

Expert Solution

Answer to Problem 1E

The first derivative of the function is $y'=−3x2−4x2$ .

Explanation of Solution

Given information:

The function $xy+2x+3x2=4$ .

Formula used:

Thechain rule for differentiation is if f is a function of gthen $ddx(f(g(x)))=f'(g(x))g'(x)$ .

Power rule for differentiation is $ddxxn=nxn−1$ .

Quotient rule for differentiation is $ddx(fg)=g(x)f'(x)−f(x)g'(x)[g(x)]2$ where f and g are functions of x .

Product rule for differentiation is $ddx(f⋅g)=f'(x)g(x)+f(x)g'(x)$ where f and g are functions of x .

Calculation:

Consider the function $xy+2x+3x2=4$ .

Isolate the value of y on left hand side on the above equation.

Subtract the terms $2x+3x2$ on both sides of the equation.

$xy=−2x−3x2+4y=−2x−3x2+4x$

Differentiate both sides with respect to x ,

$ddx(y)=ddx(−2x−3x2+4x)y'=ddx(−2x−3x2+4x)$

Recall that power rule for differentiation is $ddxxn=nxn−1$ and chain rule for differentiation is if f is a function of g then $ddx(f(g(x)))=f'(g(x))g'(x)$ also quotient rule for differentiation is $ddx(fg)=g(x)f'(x)−f(x)g'(x)[g(x)]2$ where f and g are functions of x .

Apply it. Also observe that y is a function of x,

$y'=x(−2x−3x2+4)'−(−2x−3x2+4)(x)'x2y'=x(−2−6x)−(−2x−3x2+4)1x2y'=−2x−6x2+2x+3x2−4x2y'=−3x2−4x2$

Thus, the first derivative of the function is $y'=−3x2−4x2$ .

$(c)$

To determine

To verify: The solutions obtained in parts $(a) and (b)$ are consistent.

$(c)$

Expert Solution

Explanation of Solution

Given information:

The derivative of the function $xy+2x+3x2=4$ by explicit differentiation is $y'=−3x2−4x2$ and by implicit differentiation is $y'=−y−6x−2x$ .

Consider the derivative of the function $xy+2x+3x2=4$ that is by explicit differentiation it is $y'=−3x2−4x2$ and by implicit differentiation it is $y'=−y−6x−2x$ .

In order to verify that derivative obtained implicitly and explicitly is same substitute the value of y from the function in the value of derivative found by implicit differentiation.

Consider the function $xy+2x+3x2=4$ .

Isolate the value of y on left hand side on the above equation.

Subtract the terms $2x+3x2$ on both sides of the equation.

$xy=−2x−3x2+4y=−2x−3x2+4x$

The first derivative of the function by implicit differentiationis $y'=−y−6x−2x$ .

Now,

$y'=−(−2x−3x2+4x)−6x−2x=2x+3x2−4−6x2−2xx2=−3x2−4x2$

Since, the derivative found implicitly is equal to derivative found explicitly so both the derivatives are equal.

Hence, the solutions obtained in parts $(a) and (b)$ are consistent.

Chapter 3.4, Problem 94E

Chapter 3.5, Problem 2E

Chapter 3 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 3.1 - (a) How is the number e defined? (b) Use a...Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E

Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Prob. 20E Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Prob. 26E Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Prob. 37E Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Prob. 45E Ch. 3.1 - Prob. 46E Ch. 3.1 - Prob. 47E Ch. 3.1 - Prob. 48E Ch. 3.1 - Prob. 49E Ch. 3.1 - Prob. 50E Ch. 3.1 - Prob. 51E Ch. 3.1 - Prob. 52E Ch. 3.1 - Prob. 53E Ch. 3.1 - Prob. 54E Ch. 3.1 - Prob. 55E Ch. 3.1 - Prob. 56E Ch. 3.1 - Draw a diagram to show that there are two tangent...Ch. 3.1 - Prob. 58E Ch. 3.1 - Prob. 59E Ch. 3.1 - Find the nth derivative of each function by...Ch. 3.1 - Prob. 61E Ch. 3.1 - The equation y" + y' 2y = x2 is called a...Ch. 3.1 - Prob. 63E Ch. 3.1 - Prob. 64E Ch. 3.1 - Prob. 65E Ch. 3.1 - Prob. 66E Ch. 3.1 - Prob. 67E Ch. 3.1 - Prob. 68E Ch. 3.1 - Prob. 69E Ch. 3.1 - A tangent line is drawn to the hyperbola xy = c at...Ch. 3.1 - Prob. 71E Ch. 3.1 - Prob. 72E Ch. 3.1 - Prob. 73E Ch. 3.1 - Prob. 74E Ch. 3.2 - Find the derivative of f(x) = (1 + 2x2)(x x2) in...Ch. 3.2 - Find the derivative o f the function...Ch. 3.2 - Prob. 3E Ch. 3.2 - Prob. 4E Ch. 3.2 - Differentiate. y=xex Ch. 3.2 - Differentiate. y=ex1ex Ch. 3.2 - Prob. 7E Ch. 3.2 - Prob. 8E Ch. 3.2 - Prob. 9E Ch. 3.2 - Prob. 10E Ch. 3.2 - Prob. 11E Ch. 3.2 - Prob. 12E Ch. 3.2 - Prob. 13E Ch. 3.2 - Prob. 14E Ch. 3.2 - Prob. 15E Ch. 3.2 - Prob. 16E Ch. 3.2 - Prob. 17E Ch. 3.2 - Prob. 18E Ch. 3.2 - Prob. 19E Ch. 3.2 - Prob. 20E Ch. 3.2 - Prob. 21E Ch. 3.2 - Prob. 22E Ch. 3.2 - Prob. 23E Ch. 3.2 - Prob. 24E Ch. 3.2 - 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Write the composite function in the form f(g(x))....Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 3E Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 5E Ch. 3.4 - Prob. 6E Ch. 3.4 - Prob. 7E Ch. 3.4 - Prob. 8E Ch. 3.4 - Prob. 9E Ch. 3.4 - Prob. 10E Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.4 - Prob. 13E Ch. 3.4 - Prob. 14E Ch. 3.4 - Prob. 15E Ch. 3.4 - Prob. 16E Ch. 3.4 - Prob. 17E Ch. 3.4 - Prob. 18E Ch. 3.4 - Prob. 19E Ch. 3.4 - Prob. 20E Ch. 3.4 - Prob. 21E Ch. 3.4 - Prob. 22E Ch. 3.4 - Prob. 23E Ch. 3.4 - Prob. 24E Ch. 3.4 - Prob. 25E Ch. 3.4 - Prob. 26E Ch. 3.4 - Prob. 27E Ch. 3.4 - Prob. 28E Ch. 3.4 - Prob. 29E Ch. 3.4 - Prob. 30E Ch. 3.4 - Prob. 31E Ch. 3.4 - Prob. 32E Ch. 3.4 - Prob. 33E Ch. 3.4 - Prob. 34E Ch. 3.4 - Prob. 35E Ch. 3.4 - Prob. 36E Ch. 3.4 - Prob. 37E Ch. 3.4 - Prob. 38E Ch. 3.4 - Prob. 39E Ch. 3.4 - Prob. 40E Ch. 3.4 - Prob. 41E Ch. 3.4 - Prob. 42E Ch. 3.4 - Prob. 43E Ch. 3.4 - Prob. 44E Ch. 3.4 - Prob. 45E Ch. 3.4 - Prob. 46E Ch. 3.4 - Prob. 47E Ch. 3.4 - Prob. 48E Ch. 3.4 - Prob. 49E Ch. 3.4 - At what point on the curve y=1+2x is the tangent...Ch. 3.4 - Prob. 51E Ch. 3.4 - Prob. 52E Ch. 3.4 - A table of values for f, g, f, and g is given. (a)...Ch. 3.4 - Let f and g be the functions in Exercise 63. (a)...Ch. 3.4 - Prob. 55E Ch. 3.4 - Prob. 56E Ch. 3.4 - Prob. 57E Ch. 3.4 - Prob. 58E Ch. 3.4 - Prob. 59E Ch. 3.4 - Prob. 60E Ch. 3.4 - Prob. 61E Ch. 3.4 - Prob. 62E Ch. 3.4 - Prob. 63E Ch. 3.4 - Prob. 64E Ch. 3.4 - Prob. 65E Ch. 3.4 - Prob. 66E Ch. 3.4 - Prob. 67E Ch. 3.4 - Find the 1000th derivative of f(x) = xex.Ch. 3.4 - The displacement of a particle on a vibrating...Ch. 3.4 - If the equation of motion of a particle is given...Ch. 3.4 - Prob. 71E Ch. 3.4 - Prob. 72E Ch. 3.4 - The motion of a spring that is subject to a...Ch. 3.4 - Prob. 74E Ch. 3.4 - Prob. 75E Ch. 3.4 - Prob. 76E Ch. 3.4 - Prob. 77E Ch. 3.4 - The table gives the US population from 1790 to...Ch. 3.4 - Prob. 79E Ch. 3.4 - Prob. 80E Ch. 3.4 - Prob. 81E Ch. 3.4 - Prob. 82E Ch. 3.4 - Prob. 83E Ch. 3.4 - Prob. 84E Ch. 3.4 - Prob. 85E Ch. 3.4 - Prob. 86E Ch. 3.4 - Prob. 87E Ch. 3.4 - Prob. 88E Ch. 3.4 - Prob. 89E Ch. 3.4 - Prob. 90E Ch. 3.4 - Prob. 91E Ch. 3.4 - Prob. 92E Ch. 3.4 - Prob. 93E Ch. 3.4 - Prob. 94E Ch. 3.5 - Prob. 1E Ch. 3.5 - Prob. 2E Ch. 3.5 - Prob. 3E Ch. 3.5 - Prob. 4E Ch. 3.5 - Prob. 5E Ch. 3.5 - Prob. 6E Ch. 3.5 - Prob. 7E Ch. 3.5 - Prob. 8E Ch. 3.5 - Prob. 9E Ch. 3.5 - Prob. 10E Ch. 3.5 - Prob. 11E Ch. 3.5 - Prob. 12E Ch. 3.5 - Prob. 13E Ch. 3.5 - Prob. 14E Ch. 3.5 - Prob. 15E Ch. 3.5 - Prob. 16E Ch. 3.5 - Prob. 17E Ch. 3.5 - Prob. 18E Ch. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Prob. 21E Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 23E Ch. 3.5 - Prob. 24E Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 26E Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 28E Ch. 3.5 - (a) The curve with equation y2 = 5x4 x2 is called...Ch. 3.5 - Prob. 30E Ch. 3.5 - Prob. 31E Ch. 3.5 - Prob. 32E Ch. 3.5 - Prob. 33E Ch. 3.5 - Prob. 34E Ch. 3.5 - Prob. 35E Ch. 3.5 - If x2 + xy + y3 = 1, find the value of y at the...Ch. 3.5 - Prob. 39E Ch. 3.5 - Prob. 40E Ch. 3.5 - Prob. 41E Ch. 3.5 - Prob. 42E Ch. 3.5 - Prob. 43E Ch. 3.5 - Two curves are orthogonal if their tangent lines...Ch. 3.5 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 3.5 - Prob. 46E Ch. 3.5 - Prob. 47E Ch. 3.5 - Prob. 49E Ch. 3.5 - (a) Where does the normal line to the ellipse x2 ...Ch. 3.5 - Prob. 51E Ch. 3.5 - Prob. 52E Ch. 3.5 - Prob. 53E Ch. 3.5 - Prob. 54E Ch. 3.5 - The Bessel function of order 0, y = J(x),...Ch. 3.5 - The figure shows a lamp located three units to the...Ch. 3.6 - Prob. 1E Ch. 3.6 - Prob. 2E Ch. 3.6 - Prob. 3E Ch. 3.6 - Prob. 4E Ch. 3.6 - Prob. 5E Ch. 3.6 - Prob. 6E Ch. 3.6 - Prob. 7E Ch. 3.6 - Prob. 8E Ch. 3.6 - Prob. 9E Ch. 3.6 - Prob. 10E Ch. 3.6 - Prob. 11E Ch. 3.6 - Prob. 12E Ch. 3.6 - Prob. 13E Ch. 3.6 - Prob. 14E Ch. 3.6 - Prob. 15E Ch. 3.6 - Prob. 16E Ch. 3.6 - Prob. 17E Ch. 3.6 - Prob. 18E Ch. 3.6 - Prob. 19E Ch. 3.6 - Prob. 20E Ch. 3.6 - Prob. 21E Ch. 3.6 - Prob. 22E Ch. 3.6 - Prob. 23E Ch. 3.6 - Prob. 24E Ch. 3.6 - Prob. 25E Ch. 3.6 - Prob. 26E Ch. 3.6 - Prob. 27E Ch. 3.6 - Prob. 28E Ch. 3.6 - Prob. 29E Ch. 3.6 - Prob. 30E Ch. 3.6 - Prob. 31E Ch. 3.6 - Prob. 32E Ch. 3.6 - Prob. 33E Ch. 3.6 - Prob. 34E Ch. 3.6 - Prob. 35E Ch. 3.6 - Prob. 36E Ch. 3.6 - Prob. 37E Ch. 3.6 - Prob. 38E Ch. 3.6 - Prob. 39E Ch. 3.6 - Prob. 40E Ch. 3.6 - Prob. 41E Ch. 3.6 - Prob. 42E Ch. 3.6 - Prob. 43E Ch. 3.6 - Prob. 44E Ch. 3.7 - Explain why the natural logarithmic function y =...Ch. 3.7 - Differentiate the function. f(x) = x ln x x Ch. 3.7 - Differentiate the function. f(x ) = sin(ln x)Ch. 3.7 - Differentiate the function. f(x) = ln(sin2x)Ch. 3.7 - Prob. 5E Ch. 3.7 - Prob. 6E Ch. 3.7 - Prob. 7E Ch. 3.7 - Prob. 8E Ch. 3.7 - Prob. 9E Ch. 3.7 - Prob. 10E Ch. 3.7 - Prob. 11E Ch. 3.7 - Prob. 12E Ch. 3.7 - Prob. 13E Ch. 3.7 - Prob. 14E Ch. 3.7 - Prob. 15E Ch. 3.7 - Prob. 16E Ch. 3.7 - Prob. 17E Ch. 3.7 - Prob. 18E Ch. 3.7 - Prob. 19E Ch. 3.7 - Prob. 20E Ch. 3.7 - Prob. 21E Ch. 3.7 - Prob. 22E Ch. 3.7 - Prob. 23E Ch. 3.7 - Prob. 24E Ch. 3.7 - 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Prob. 20E Ch. 3.8 - Prob. 21E Ch. 3.8 - Prob. 22E Ch. 3.8 - Prob. 23E Ch. 3.8 - Prob. 24E Ch. 3.8 - The table shows how the average age of first...Ch. 3.8 - Refer to the law of laminar flow given in Example...Ch. 3.8 - Prob. 28E Ch. 3.8 - Prob. 29E Ch. 3.8 - The cost function for a certain commodity is C(q)...Ch. 3.8 - Prob. 31E Ch. 3.8 - Prob. 32E Ch. 3.8 - Patients undergo dialysis treatment to remove urea...Ch. 3.8 - Invasive species often display a wave of advance...Ch. 3.8 - Prob. 35E Ch. 3.9 - Prob. 1E Ch. 3.9 - Prob. 2E Ch. 3.9 - Prob. 3E Ch. 3.9 - Prob. 4E Ch. 3.9 - Prob. 5E Ch. 3.9 - Prob. 6E Ch. 3.9 - Prob. 7E Ch. 3.9 - Prob. 8E Ch. 3.9 - Prob. 9E Ch. 3.9 - Prob. 10E Ch. 3.9 - Prob. 11E Ch. 3.9 - Prob. 12E Ch. 3.9 - Prob. 13E Ch. 3.9 - Prob. 14E Ch. 3.9 - Prob. 15E Ch. 3.9 - Prob. 16E Ch. 3.9 - Prob. 17E Ch. 3.9 - Prob. 18E Ch. 3.9 - Prob. 19E Ch. 3.9 - Prob. 20E Ch. 3.9 - Prob. 21E Ch. 3.9 - Prob. 22E Ch. 3.9 - Prob. 23E Ch. 3.9 - Prob. 24E Ch. 3.9 - Prob. 25E Ch. 3.9 - Prob. 26E Ch. 3.9 - Prob. 27E Ch. 3.9 - Prob. 28E Ch. 3.9 - The circumference of a sphere was measured to be...Ch. 3.9 - Use differentials to estimate the amount of paint...Ch. 3.9 - Prob. 31E Ch. 3.9 - Prob. 32E Ch. 3.9 - Prob. 33E Ch. 3.9 - Prob. 34E Ch. 3.9 - Prob. 35E Ch. 3.9 - Prob. 36E Ch. 3 - State each differentiation rule both in symbols...Ch. 3 - Prob. 2RCC Ch. 3 - Prob. 3RCC Ch. 3 - Prob. 4RCC Ch. 3 - Prob. 5RCC Ch. 3 - Prob. 6RCC Ch. 3 - Prob. 1RQ Ch. 3 - Prob. 2RQ Ch. 3 - Prob. 3RQ Ch. 3 - Prob. 4RQ Ch. 3 - Prob. 5RQ Ch. 3 - Prob. 6RQ Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 8RQ Ch. 3 - Prob. 9RQ Ch. 3 - Prob. 10RQ Ch. 3 - Prob. 11RQ Ch. 3 - Prob. 12RQ Ch. 3 - Prob. 1RE Ch. 3 - Prob. 2RE Ch. 3 - Prob. 3RE Ch. 3 - Prob. 4RE Ch. 3 - Prob. 5RE Ch. 3 - Prob. 6RE Ch. 3 - Prob. 7RE Ch. 3 - Prob. 8RE Ch. 3 - Prob. 9RE Ch. 3 - Prob. 10RE Ch. 3 - Prob. 11RE Ch. 3 - Prob. 12RE Ch. 3 - Prob. 13RE Ch. 3 - Prob. 14RE Ch. 3 - Prob. 15RE Ch. 3 - Prob. 16RE Ch. 3 - Prob. 17RE Ch. 3 - 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The first derivative of the function x y + 2 x + 3 x 2 = 4 by implicit differentiation .

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To calculate: The first derivative of the function xy+2x+3x2=4<m:math><m:mrow><m:mi>x</m:mi><m:mi>y</m:mi><m:mo>+</m:mo><m:mn>2</m:mn><m:mi>x</m:mi><m:mo>+</m:mo><m:mn>3</m:mn><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>=</m:mo><m:mn>4</m:mn></m:mrow></m:math> by implicit differentiation.

Answer to Problem 1E

Explanation of Solution

To calculate: The first derivative of the function xy+2x+3x2=4<m:math><m:mrow><m:mi>x</m:mi><m:mi>y</m:mi><m:mo>+</m:mo><m:mn>2</m:mn><m:mi>x</m:mi><m:mo>+</m:mo><m:mn>3</m:mn><m:msup><m:mi>x</m:mi><m:mn>2</m:mn></m:msup><m:mo>=</m:mo><m:mn>4</m:mn></m:mrow></m:math> by explicit differentiation.

Answer to Problem 1E

Explanation of Solution

To verify: The solutions obtained in parts (a) and (b)<m:math><m:mrow><m:mrow><m:mo>(</m:mo><m:mi>a</m:mi><m:mo>)</m:mo></m:mrow><m:mtext> and </m:mtext><m:mrow><m:mo>(</m:mo><m:mi>b</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> are consistent.

Explanation of Solution

Chapter 3 Solutions

To calculate: The first derivative of the function $xy+2x+3x2=4$ by implicit differentiation.

To calculate: The first derivative of the function $xy+2x+3x2=4$ by explicit differentiation.

To verify: The solutions obtained in parts $(a) and (b)$ are consistent.