Textbook Question
Use differentiation to verify each equation.
d/dx(x / √1−x²) = 1 / (1−x²)^3/2.
3. Techniques of Differentiation
Product and Quotient Rules
5:14 minutesProblem 3.R.84b
Textbook Question
Use the given graphs of f and g to find each derivative. <IMAGE>
b. d/dx (f(x)g(x)) |x=1
Verified step by step guidance1
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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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 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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