Use the given graphs of f and g to find each derivative. b. d/dx (f(x)g(x)) |x=1

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.R.84b

Capitolo 3, Problema 3.R.84b

Use the given graphs of f and g to find each derivative. <IMAGE>
b. d/dx (f(x)g(x)) |x=1

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To find the derivative of the product of two functions, f(x) and g(x), at a specific point, we use the product rule. The product rule states that the derivative of a product of two functions is given by: (d/dx)[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

Evaluate the derivative at x = 1. This means we need to find f'(1), g(1), f(1), and g'(1) from the given graphs.

First, find f'(1) from the graph of f. This is the slope of the tangent line to the curve of f at x = 1.

Next, find g(1) from the graph of g. This is the value of the function g at x = 1.

Similarly, find f(1) from the graph of f and g'(1) from the graph of g. Use these values in the product rule formula to find the derivative at x = 1: f'(1)g(1) + f(1)g'(1).

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Product Rule

The Product Rule is a fundamental differentiation rule used to find the derivative of the product of two functions. It states that if you have two functions, f(x) and g(x), the derivative of their product is given by d/dx(f(x)g(x)) = f'(x)g(x) + f(x)g'(x). This rule is essential for solving problems involving the multiplication of functions.

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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, you will need to compute the derivatives of f and g at x=1, and then apply the Product Rule to find the derivative of their product at that point. This step is crucial for obtaining the final answer.

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Graph Interpretation

Graph interpretation is the ability to analyze and extract information from graphical representations of functions. In this context, understanding the graphs of f and g is vital for determining their values and slopes at x=1. This skill helps in visualizing how the functions behave and aids in accurately applying the Product Rule.

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