Textbook Question
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(s) = 4s³+3s; a= -3, -1
3. Techniques of Differentiation
Basic Rules of Differentiation
2:44 minutesProblem 12
Textbook Question
Use the table to find the following derivatives.
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d/dx (f(x) + g(x)) ∣x=1
Verified step by step guidance1
Step 1: Recall the sum rule for derivatives, which states that the derivative of a sum of functions is the sum of their derivatives. In mathematical terms, this is expressed as \( \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \).
Step 2: Identify the point at which you need to evaluate the derivative, which is \( x = 1 \) in this problem.
Step 3: Use the table provided in the problem to find the values of \( f'(1) \) and \( g'(1) \). These are the derivatives of \( f(x) \) and \( g(x) \) evaluated at \( x = 1 \).
Step 4: Substitute the values of \( f'(1) \) and \( g'(1) \) from the table into the expression \( f'(1) + g'(1) \).
Step 5: Simplify the expression to find the value of the derivative \( \frac{d}{dx} (f(x) + g(x)) \) at \( x = 1 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
A derivative represents the rate at which a function changes at a given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides crucial information about the function's behavior, such as its slope and concavity.
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Sum Rule of Derivatives
The Sum Rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if f(x) and g(x) are differentiable functions, then d/dx (f(x) + g(x)) = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with the addition of functions.
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Evaluating Derivatives at a Point
Evaluating a derivative at a specific point involves substituting the value of that point into the derivative function. For example, to find d/dx (f(x) + g(x)) at x=1, you first compute f'(1) and g'(1) using the derivatives obtained from the Sum Rule, and then add these values together to get the final result.
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