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.

Full solution

Q. 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.

Identify Function Components: Identify the function and its components. $\newline$ 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. $\newline$ 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. $\newline$ 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. $\newline$ 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: $\newline$ $\frac{dy}{dx} = -\csc(-7x - 2)\cot(-7x - 2)(-7)$ .
Simplify Expression: Simplify the expression. $\newline$ We can simplify the expression by multiplying through by $-7$ : $\newline$ $\frac{dy}{dx} = 7\csc(-7x - 2)\cot(-7x - 2)$ .