We are given the function Y = log [ a bsinx / absinx ].

Using the logarithmic differentiation formula, we can find the derivative of Y with respect to x as:

dY/dx = [1 / ( a bsinx / absinx ) ] * [ d / dx ( a bsinx / absinx ) ]



Now, we need to simplify the expression inside the derivative.

a and b are constants, so we can take them outside the derivative:

dY/dx = [1 / ( a bsinx / absinx ) ] * [ a b * d / dx (sinx / sinx) ]

Now, we can simplify the expression inside the derivative as follows:

sinx / sinx = 1

So, the derivative becomes:

dY/dx = [1 / ( a bsinx / absinx ) ] * [ a b * d / dx (1) ]



The derivative of 1 with respect to x is 0, so the expression inside the derivative becomes 0:

dY/dx = [1 / ( a bsinx / absinx ) ] * 0

dY/dx = 0



The derivative of Y with respect to x is 0. This means that Y is a constant with respect to x. Therefore, the graph of Y is a horizontal line, and there is no slope at any point.

In other words, the function Y does not change as x changes. This result may seem surprising, but it is a consequence of the way the function is defined.



dy/dx = 0.