Question

find derivative of y=cosec(5x)

Answer 1

Here is the required answer

Answer 2

There are six trigonometric ratios and one of them is cosecant or abbreviated as $csc\,\theta$ or $cosec\,\theta$ .

Things to remember about $\bf{cosec\,\theta}$

• $cosec\,\theta = \frac{hypotenuse}{opposite\,\, side}$

• $cosec\,\theta$ is the reciprocal of $sin\,\theta$

        That is,   $cosec\,\theta = \frac{1}{sin\,\theta}$

To find: The derivative of y = cosec(5x)

Let us find the derivative of cosec(x),

We know that,

               $cosec\,x = \frac{1}{sin\,x} = (sinx)^{-1}$

Differentiating the above equation, we get,

            $\frac{d(cosec(ax))}{dx} =\frac{d(sin(ax))^{-1}}{dx} = -\,(sin(ax))^{-2}\times\frac{d(sin(ax))}{dx}$

                                                    $=-(sin(ax))^{-2}\times a cos(ax)\\\\=-a\,\frac{cos(ax)}{sin(ax)} \times \frac{1}{sin(ax)}\\\\ =-a\,cot(ax) \,\,cosec(ax)$

∴      $\bf\frac{d(cosec(ax))}{dx} = -a\,cot(ax)\,\,cosec(ax)$

⇒ The derivative of y = cosec(5x) is,

That is,   $\frac{d(cosec(5x))}{dx} = -5\,\,cot(5x)\,\,cosec(5x)$

                         

Answer:       $\bf\frac{d(cosec(5x))}{dx} = -5\,\,cot(5x)\,\,cosec(5x)$

             

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