9–61. Evaluate and simplify y'.


y = csc⁵ 3x

Verified step by step guidance

1. First, recognize that the problem involves differentiating a function. We need to find the derivative of y with respect to x, denoted as y'.
2. The given function is y = csc⁵(3x). To differentiate this, apply the chain rule. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).
3. Identify the outer function as f(u) = u⁵, where u = csc(3x). The derivative of u⁵ with respect to u is 5u⁴.
4. Next, differentiate the inner function u = csc(3x). The derivative of csc(x) is -csc(x)cot(x), so the derivative of csc(3x) is -csc(3x)cot(3x) * 3, using the chain rule again.
5. Combine these results using the chain rule: y' = 5(csc(3x))⁴ * (-csc(3x)cot(3x) * 3). Simplify the expression by multiplying the constants and combining like terms.

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 differentiate the function y = csc⁵(3x) 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 evaluated at the inner function, multiplied by the derivative of the inner function. For y = csc⁵(3x), applying the Chain Rule is essential to correctly differentiate the function.

Trigonometric Derivatives

Trigonometric derivatives are the derivatives of trigonometric functions, which have specific rules. For example, the derivative of csc(x) is -csc(x)cot(x). Understanding these rules is crucial for differentiating y = csc⁵(3x), as we will need to apply the derivative of the csc function in our calculations.