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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 92a
Chapter 3, Problem 92a

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)

Verified step by step guidance
1
Step 1: Identify the given information. We have two functions, f(x) and g(x), with their respective tangent lines at x=2. The tangent line to f(x) at x=2 is y=4x+1, which implies that f'(2)=4 and f(2) is the y-value when x=2, which can be found by substituting x=2 into the equation of the tangent line. Similarly, the tangent line to g(x) at x=2 is y=3x−2, which implies that g'(2)=3 and g(2) is the y-value when x=2, found by substituting x=2 into the equation of the tangent line.
Step 2: Use the product rule to find the derivative of y=f(x)g(x). The product rule states that if y=u(x)v(x), then y'=u'(x)v(x) + u(x)v'(x). Here, u(x)=f(x) and v(x)=g(x), so y'=f'(x)g(x) + f(x)g'(x).
Step 3: Substitute x=2 into the derivative found in Step 2. This gives y'(2)=f'(2)g(2) + f(2)g'(2).
Step 4: Substitute the known values from Step 1 into the expression from Step 3. We have f'(2)=4, g'(2)=3, and the values of f(2) and g(2) can be found from the tangent line equations.
Step 5: Use the result from Step 4 to write the equation of the tangent line to y=f(x)g(x) at x=2. The equation of a tangent line is y-y_1=m(x-x_1), where m is the slope found in Step 4, and (x_1, y_1) is the point on the curve at x=2, which can be found by evaluating y=f(x)g(x) at x=2.

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Key Concepts

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

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the derivative of the function at that point, indicating the rate of change of the function. In this problem, the equations of the tangent lines for functions f and g at x=2 provide the necessary slopes for further calculations.
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Slopes of Tangent Lines

Product Rule

The Product Rule is a fundamental differentiation rule used when finding the derivative of the product of two functions. It states that if u(x) and v(x) are two differentiable functions, then the derivative of their product is given by u'v + uv'. This rule is essential for determining the derivative of the product of f(x) and g(x) at x=2 in the given problem.
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The Product Rule

Chain Rule

The Chain Rule is a method for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. While not directly applied in this problem, understanding the Chain Rule is important for more complex scenarios involving nested functions.
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Intro to the Chain Rule
Related Practice
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

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

a. (f^-1)'(7)

1
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Textbook Question

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)

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)