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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 92b
Chapter 3, Problem 92b

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.
b. y = f(x) / g(x)

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To find the equation of the tangent line to the curve y = f(x) / g(x) at x = 2, we need to use the quotient rule for derivatives. The quotient rule states that if y = u(x) / v(x), then the derivative y' is given by (v(x)u'(x) - u(x)v'(x)) / (v(x))^2.
In this problem, u(x) = f(x) and v(x) = g(x). We know the derivatives at x = 2 from the tangent lines: f'(2) = 4 and g'(2) = 3.
Apply the quotient rule: y' = (g(x)f'(x) - f(x)g'(x)) / (g(x))^2. Substitute x = 2, f'(2) = 4, and g'(2) = 3 into this formula.
Calculate the values of f(2) and g(2) using the tangent line equations. For f(x), the tangent line is y = 4x + 1, so f(2) = 4(2) + 1. For g(x), the tangent line is y = 3x - 2, so g(2) = 3(2) - 2.
Substitute f(2), g(2), f'(2), and g'(2) into the quotient rule formula to find y'(2). This y'(2) is the slope of the tangent line to y = f(x) / g(x) at x = 2. Use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the curve at x = 2, to write the equation of the tangent line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is represented as the slope of the tangent line to the graph of the function at that point. Understanding derivatives is crucial for analyzing the behavior of functions and finding tangent lines.
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Derivatives

Quotient Rule

The quotient rule is a formula used to find the derivative of a function that is the ratio of two other functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient is given by (u/v)' = (u'v - uv')/v². This rule is essential for differentiating the function y = f(x) / g(x) in the given problem.
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The Quotient Rule

Tangent Line Equation

The equation of a tangent line at a point on a curve can be expressed in the form y = mx + b, where m is the slope (the derivative at that point) and b is the y-intercept. To find the tangent line for the function y = f(x) / g(x) at x=2, one must calculate the derivative at that point and use the point-slope form of the line.
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Equations of Tangent Lines
Related Practice
Textbook Question

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

y = f(x)g(x)

Textbook Question

If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>

b. (f^-1)'(3)

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Textbook Question

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.


a. Find L′(1) and interpret the meaning of this value.

Textbook Question

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

Textbook Question

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Textbook Question

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(1)