Find the derivative of y=csc^(-7x-2)(x). Be sure to include parentheses around the arguments of any logarithmic or trigonometric functions in your answer.

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Find the derivative of  y=csc^(-7x-2)(x). Be sure to include parentheses around the arguments of any logarithmic or trigonometric functions in your answer.

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Identify Function Components: Identify the function and its components. We have the function y = csc^(-7x - 2)(x), which is a composite function involving the cosecant function and a linear function inside it. We will need to use the chain rule to find the derivative.Apply Chain Rule: Apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is csc(u) and the inner function is u = -7x - 2.Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of csc(u) with respect to u is -csc(u)cot(u). We will apply this to our function, keeping in mind that u = -7x - 2.Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of u = -7x - 2 with respect to x is -7.Combine Derivatives: Combine the derivatives using the chain rule. The derivative of y with respect to x is the derivative of the outer function times the derivative of the inner function, which gives us: dy/dx = -csc(-7x - 2)cot(-7x - 2)(-7).Simplify Expression: Simplify the expression. We can simplify the expression by multiplying through by -7: dy/dx = 7csc(-7x - 2)cot(-7x - 2).

Find the derivative of $y=\csc ^{-7 x-2}(x)$ . Be sure to include parentheses around the arguments of any logarithmic or trigonometric functions in your answer.

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Find the derivative of y=csc⁡−7x−2(x) y=\csc ^{-7 x-2}(x) y=csc−7x−2(x). Be sure to include parentheses around the arguments of any logarithmic or trigonometric functions in your answer.

Find the derivative of $y=\csc ^{-7 x-2}(x)$ . Be sure to include parentheses around the arguments of any logarithmic or trigonometric functions in your answer.