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Solve the equation.

StartFraction d Over d x EndFraction cosecant Superscript negative 1 Baseline x squared =

StartFraction negative 2 x Over StartRoot 1 minus x Superscript 4 Baseline EndRoot EndFraction

StartFraction negative 2 Over x StartRoot x Superscript 4 Baseline minus 1 EndRoot EndFraction

StartFraction 2 Over x StartRoot x Superscript 4 Baseline minus 1 EndRoot EndFraction

StartFraction 2 x Over StartRoot 1 minus x Superscript 4 Baseline EndRoot EndFraction

User Magnetar
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1 Answer

2 votes

The solution to the equation
(d)/(dx)(\csc^(-1)(x^2)) is
(dy)/(dx) = (2)/(-x\cot(\csc^(-1)(x^2))\cdot)

How to determine the solution to the equation

From the question, we have the following parameters that can be used in our computation:


(d)/(dx)(\csc^(-1)(x^2))

This can be expressed as


y = \csc^(-1)(x^2)

Take the cosecant of both sides

So, we have


\csc(y) = x^2

Next, we differentiate both sides


-\cot(y)\csc(y) (dy)/(dx) = 2x

Make dy/dx the subject


(dy)/(dx) = (2x)/(-\cot(y)\csc(y) )

Recall that
y = \csc^(-1)(x^2) and
\csc(y) = x^2

So, we have


(dy)/(dx) = (2x)/(-\cot(\csc^(-1)(x^2))\cdot x^2 )

Divide


(dy)/(dx) = (2)/(-x\cot(\csc^(-1)(x^2))\cdot)

Hence, the solution to the equation is
(dy)/(dx) = (2)/(-x\cot(\csc^(-1)(x^2))\cdot)

Question

Solve the equation
(d)/(dx)(\csc^(-1)(x^2))

User Chris Beach
by
8.9k points