Textbook Question
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
d. d/dx (f(x)³) |x=5
3. Techniques of Differentiation
The Chain Rule
4:43 minutesProblem 3.38
Textbook Question
9–61. Evaluate and simplify y'.
y = (v / v+1)^4/3
Verified step by step guidance1
Step 1: Identify the function y = \left(\frac{v}{v+1}\right)^{\frac{4}{3}} and recognize that it is a composite function, which suggests the use of the chain rule for differentiation.
Step 2: Apply the chain rule. The chain rule states that if you have a composite function y = f(g(v)), then the derivative y' = f'(g(v)) \cdot g'(v). Here, let u = \frac{v}{v+1}, so y = u^{\frac{4}{3}}.
Step 3: Differentiate the outer function with respect to u. The derivative of u^{\frac{4}{3}} with respect to u is \frac{4}{3}u^{\frac{1}{3}}.
Step 4: Differentiate the inner function u = \frac{v}{v+1} with respect to v. Use the quotient rule: if u = \frac{a}{b}, then u' = \frac{a'b - ab'}{b^2}. Here, a = v and b = v+1, so a' = 1 and b' = 1.
Step 5: Combine the results from Steps 3 and 4 using the chain rule. Multiply the derivative of the outer function by the derivative of the inner function to find y'.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this case, we need to apply differentiation rules to the given function y to find y'.
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Chain Rule
The Chain Rule is a technique used in differentiation when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. This rule will be essential for differentiating the function y = (v / (v + 1))^(4/3) since it involves a power and a quotient.
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Quotient Rule
The Quotient Rule is a specific rule for differentiating functions that are expressed as the ratio of two other functions. It states that if you have a function in the form of f(v) = g(v) / h(v), the derivative f'(v) is given by (g'(v)h(v) - g(v)h'(v)) / (h(v))^2. This rule will be necessary to differentiate the function y, which is a quotient of two expressions.
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