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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 3.4.89
Chapter 3, Problem 3.4.89

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>
d/dx (xg(x)) | x=2

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Step 1: Recognize that you need to find the derivative of the product of two functions, x and g(x), at x = 2. This requires using the product rule for differentiation.
Step 2: Recall the product rule formula: if you have two functions u(x) and v(x), then the derivative of their product is given by (u(x)v(x))' = u'(x)v(x) + u(x)v'(x).
Step 3: Identify u(x) = x and v(x) = g(x). Compute the derivatives: u'(x) = 1 and v'(x) = g'(x).
Step 4: Apply the product rule: (xg(x))' = 1 * g(x) + x * g'(x).
Step 5: Evaluate the expression at x = 2: substitute x = 2 into the expression 1 * g(x) + x * g'(x) to find the derivative at that point. Use the graph to find the values of g(2) and g'(2).

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Key Concepts

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

Product Rule

The Product Rule is a fundamental differentiation technique used when finding the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x). This rule is essential for solving problems involving products of functions, such as xg(x) in the given question.
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Evaluating Derivatives

Evaluating derivatives involves substituting a specific value into the derivative function to find the slope of the tangent line at that point. In this case, after applying the Product Rule to find d/dx (xg(x)), you will substitute x = 2 into the resulting expression. This step is crucial for determining the instantaneous rate of change of the function at that specific point.
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Function Notation

Function notation is a way to represent functions and their derivatives clearly. In the expression g(x), g represents a function of x, and its derivative is denoted as g'(x). Understanding function notation is vital for interpreting the problem correctly, as it allows you to identify the functions involved and apply the appropriate differentiation rules effectively.
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Related Practice
Textbook Question

Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)

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Textbook Question

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x²(3y²−2y³) = 4

Textbook Question

The speed of sound (in m/s) in dry air is approximated the function v(T) = 331 + 0.6T, where T is the air temperature (in degrees Celsius). Evaluate v' (T) and interpret its meaning.

Textbook Question

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

An object dropped from rest falls d(t)=16t² feet in t seconds. Find d′(4).

Textbook Question

Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

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Textbook Question

9–61. Evaluate and simplify y'.

y = (v / v+1)^4/3