Textbook Question
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 1/√t; a=9, 1/4
3. Techniques of Differentiation
Basic Rules of Differentiation
5:53 minutesProblem 11
Textbook Question
Let F(x) = f(x) + g(x),G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x), where the graphs of f and g are shown in the figure. Find each of the following.
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H'(2)
Verified step by step guidance1
Step 1: Understand the problem. We are given three functions: F(x) = f(x) + g(x), G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x). We need to find the derivative of H at x = 2, denoted as H'(2).
Step 2: Use the linearity of differentiation. The derivative of a sum is the sum of the derivatives, and the derivative of a constant times a function is the constant times the derivative of the function. Therefore, H'(x) = 3f'(x) + 2g'(x).
Step 3: Evaluate the derivatives of f and g at x = 2. From the graph, determine the values of f'(2) and g'(2). These are the slopes of the tangent lines to the graphs of f and g at x = 2.
Step 4: Substitute the values of f'(2) and g'(2) into the expression for H'(x). This gives H'(2) = 3f'(2) + 2g'(2).
Step 5: Calculate H'(2) using the values obtained from the graph. This will give you the rate of change of H at x = 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the graph of the function at a given point. For a function F(x), the derivative F'(x) can be found using rules such as the sum, difference, and constant multiple rules.
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Sum and Difference of Functions
When dealing with the sum or difference of two functions, the derivative can be computed by applying the sum and difference rules. Specifically, if F(x) = f(x) + g(x), then F'(x) = f'(x) + g'(x). Similarly, for G(x) = f(x) - g(x), G'(x) = f'(x) - g'(x). This property allows for the straightforward calculation of derivatives for combined functions.
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Linear Combination of Functions
A linear combination of functions involves multiplying each function by a constant and then adding the results. In the case of H(x) = 3f(x) + 2g(x), the coefficients 3 and 2 indicate how much each function contributes to H. The derivative of H can be found by applying the constant multiple rule, which states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
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