Textbook Question
Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
c. (f^-1)'(f(2))
3. Techniques of Differentiation
Basic Rules of Differentiation
5:23 minutesProblem 3.2.27b
Textbook Question
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 1/√t; a=9, 1/4
Verified step by step guidance1
Step 1: Identify the function f(t) = \frac{1}{\sqrt{t}} and recognize that you need to find its derivative, f'(t).
Step 2: Rewrite the function in a form that is easier to differentiate: f(t) = t^{-1/2}.
Step 3: Use the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n \cdot t^{n-1}. Apply this to f(t) = t^{-1/2}.
Step 4: Calculate the derivative: f'(t) = -\frac{1}{2} \cdot t^{-3/2}.
Step 5: Evaluate f'(t) at the given values of a. First, substitute a = 9 into f'(t) to find f'(9), and then substitute a = \frac{1}{4} into f'(t) to find f'(\frac{1}{4}).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that involve square roots or other composite forms.
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Evaluating Derivatives at Specific Points
To evaluate the derivative at a specific point, you first need to find the derivative function and then substitute the given value into this function. This process allows you to determine the instantaneous rate of change of the original function at that particular point. In this case, you will compute f'(9) and f'(1/4) for the function f(t) = 1/√t.
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