9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

Verified step by step guidance

First, recognize that the expression y = (sin x / (cos x + 1))^(1/3) is a composite function, which involves the chain rule for differentiation.

Identify the outer function as u^(1/3) where u = sin x / (cos x + 1). The derivative of u^(1/3) with respect to u is (1/3)u^(-2/3).

Next, differentiate the inner function u = sin x / (cos x + 1) using the quotient rule. The quotient rule states that if u = f(x)/g(x), then u' = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.

Apply the quotient rule: Let f(x) = sin x and g(x) = cos x + 1. Then f'(x) = cos x and g'(x) = -sin x. Substitute these into the quotient rule formula.

Finally, combine the derivatives using the chain rule: y' = (1/3)(sin x / (cos x + 1))^(-2/3) * ((cos x)(cos x + 1) - (sin x)(-sin x)) / (cos x + 1)^2. Simplify the expression to find y'.

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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 how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. In this context, we need to apply the rules of differentiation to find y', the derivative of the given function y.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is particularly relevant here since y is expressed as a function raised to a power, requiring the application of the chain rule.

Quotient Rule

The quotient rule is used to differentiate functions that are expressed as the ratio of two other functions. It states that if y = f(x)/g(x), then the derivative y' is given by (f'g - fg')/g². In this problem, since y involves a fraction (sin x / (cos x + 1)), applying the quotient rule will be necessary to correctly find the derivative.

9–61. Evaluate and simplify y'.y = (sin x / cos x+1)^1/3