Textbook Question
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(t) = 1/t+1; a=1
3. Techniques of Differentiation
Product and Quotient Rules
2:10 minutesProblem 3.86
Textbook Question
Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).
Verified step by step guidance1
Step 1: Identify the function y = x f(x) and recognize that you need to find the derivative of this function to determine the slope of the curve at a specific point.
Step 2: Use the product rule to differentiate y = x f(x). The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is given by (uv)' = u'v + uv'.
Step 3: Apply the product rule to y = x f(x). Here, u(x) = x and v(x) = f(x). Therefore, the derivative y' = (x)'f(x) + x(f(x))'.
Step 4: Calculate the derivatives: (x)' = 1 and (f(x))' = f'(x). Substitute these into the expression from Step 3 to get y' = 1*f(x) + x*f'(x).
Step 5: Substitute the given values f(1) = 2 and f'(1) = 2 into the derivative expression y' = f(x) + x*f'(x) to find the slope at the point (1, 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 a 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 u'v + uv'. This rule is essential for differentiating the function y = x f(x) in the given problem.
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The Product Rule
Derivative
A derivative represents the rate of change of a function with respect to its variable. It provides information about the slope of the tangent line to the curve at any given point. In this context, calculating the derivative of y = x f(x) will allow us to find the slope of the curve at the specific point (1, 2).
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Evaluating Derivatives at a Point
Evaluating derivatives at a specific point involves substituting the x-value of that point into the derivative function. This process yields the slope of the tangent line to the curve at that point. In this case, after applying the Product Rule and finding the derivative, we will substitute x = 1 to determine the slope of the curve y = x f(x) at (1, 2).
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