Question

Derivative of (csc^-1u)

Answer 1

The derivative of \(csc^{-1}(u)\) is \(-\frac{1}{u\sqrt{u^2 - 1}}\).

Step-by-step explanation:

The derivative of the cosecant inverse function \( csc^{-1}(u) \) with respect to u can be found using implicit differentiation. Assume y is the angle whose cosecant is u, so u = csc(y). Differentiating both sides with respect to u gives \1 = -\csc(y)\cot(y)\frac{dy}{du}\, leading to \\frac{dy}{du} = -\frac{1}{\csc(y)\cot(y)}\.

Rewriting csc and cot in terms of sine and cosine, and knowing that sin(y) = 1/u, we get \\cot(y) = \sqrt{u^2 - 1}\. Therefore, the derivative is \\frac{dy}{du} = -\frac{1}{\sqrt{u^2 - 1}}\cdot\frac{1}{u}\, and after simplifying this equals \-\frac{1}{u\sqrt{u^2 - 1}}\.

The derivative of (csc^-1u) can be found using the chain rule. Let's break it down:

Start with the derivative of the inverse cosecant function, which is (1/|u|)

Then, multiply it by the derivative of u with respect to x. Let's call this derivative du/dx.

Finally, simplify the expression by combining the two parts: (1/|u|) * (du/dx)

So, the derivative of (csc^-1u) is (1/|u|) * (du/dx).