Given y = cos(3x)cosec(5x). Find dy/dx.

To find dy/dx, we need to differentiate the given equation y = cos(3x)cosec(5x) with respect to x using the rules of differentiation. Let’s go step by step:

First, rewrite cosec(5x) as 1/sin(5x). So, we have:

y = cos(3x)(1/sin(5x))

Now, using the product rule, the derivative of cos(3x)(1/sin(5x)) with respect to x is:

dy/dx = (d/dx[cos(3x)])(1/sin(5x)) + cos(3x)(d/dx[1/sin(5x)])

The derivative of cos(3x) with respect to x is -3sin(3x), and the derivative of 1/sin(5x) can be calculated using the chain rule as -cos(5x)/(sin^2(5x)). So, we have:

dy/dx = (-3sin(3x))(1/sin(5x)) + cos(3x)(-cos(5x)/(sin^2(5x)))

Simplifying further, we get:

dy/dx = -3sin(3x)/sin(5x) – cos(3x)cos(5x)/(sin^2(5x))

Therefore, dy/dx is equal to:

dy/dx = -3sin(3x)/sin(5x) – cos(3x)cos(5x)/(sin^2(5x))

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