9–61. Evaluate and simplify y'.
y = (v / v+1)^4/3

Verified step by step guidance

1. 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.
2. 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^{\(\frac{4}{3}\)}.
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^{\(\frac{1}{3}\)}.
4. 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.
5. 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'.

Key Concepts

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'.

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.

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.