The derivative of cosec x² is [tex]-2x.cot \ x^{2} .cosec \ x^{2}[/tex]
Differentiation
From the question, we are to differentiate cosec x²
Let y = cosec x², then we will determine the value of [tex]\frac{dy}{dx}[/tex]
[tex]y = cosec \ x^{2}[/tex]
Let [tex]u = x^{2}[/tex]
Then,
[tex]\frac{du}{dx}=2x[/tex]
Also, [tex]y = cosec \ x^{2}[/tex] becomes
[tex]y = cosec \ u[/tex]
Then,
[tex]\frac{dy}{du} = - cot \ u . cosec \ u[/tex]
From Chain rule
[tex]\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}[/tex]
∴ [tex]\frac{dy}{dx} = -cot \ u.cosec \ u \times 2x[/tex]
Recall that [tex]u = x^{2}[/tex]
Then,
[tex]\frac{dy}{dx} = -cot \ x^{2} .cosec \ x^{2} \times 2x[/tex]
[tex]\frac{dy}{dx} = -2x.cot \ x^{2} .cosec \ x^{2}[/tex]
Hence, the derivative of cosec x² is [tex]-2x.cot \ x^{2} .cosec \ x^{2}[/tex]
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