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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 92b
Capitolo 3, Problema 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)

Guida verificata passo dopo passo
1
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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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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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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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
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