Question

Find dy/dx if y=csc-1(e^x)
ind \frac{d y}{d x} if y=\csc ^{-1}(e^{x}) \frac{d y}{d x}=

Answer 1

To find the derivative dy/dx of y = csc-1(e^x), we use the chain rule along with the knowledge of the derivative of inverse cosecant function, resulting in dy/dx = -1 / sqrt(e^(2x) - 1).

Step-by-step explanation:

The question involves finding the derivative dy/dx when y is equal to the inverse cosecant of e to the power of x, which is written as y = csc-1(ex). To find the derivative of an inverse trigonometric function, we can use implicit differentiation.

Let's start by taking the derivative of both sides with respect to x:

dy/dx = d/dx [csc-1(ex)].

We know that the derivative of an inverse cosecant function csc-1u with respect to u is -1/(|u| sqrt(u2 - 1)), and the derivative of ex is ex.

Applying the chain rule, we get:

dy/dx = -ex / (|ex| sqrt(e2x - 1))

Since ex is always positive, we can simplify the expression:

dy/dx = -ex / (ex sqrt(e2x - 1)) = -1 / sqrt(e2x - 1).