Differentiate (3sinx)/(2sinx+2cosx)

To differentiate the function (3sinx)/(2sinx+2cosx), we will use the quotient rule for differentiation which states that the derivative of two functions f(x) and g(x) is given by:

(f'(x)g(x) - f(x)g'(x)) / [g(x)]^2

Here, our f(x) is 3sinx and g(x) is 2sinx+2cosx.

Step 1: Differentiate f(x) = 3sinx The derivative of sinx is cosx, so f'(x) = 3cosx.

Step 2: Differentiate g(x) = 2sinx+2cosx The derivative of sinx is cosx and the derivative of cosx is -sinx, so g'(x) = 2cosx - 2sinx.

Step 3: Substitute f(x), f'(x), g(x), and g'(x) into the quotient rule formula This gives us: [(3cosx)(2sinx+2cosx) - (3sinx)(2cosx - 2sinx)] / (2sinx+2cosx)^2

Step 4: Simplify the expression This simplifies to: [6cos^2x + 6sinxcosx - 6sinxcosx + 6sin^2x] / (4sin^2x + 8sinxcosx + 4cos^2x)

Step 5: Further simplify the expression This simplifies to: [6(cos^2x + sin^2x)] / [4(sin^2x + 2sinxcosx + cos^2x)]

Since sin^2x + cos^2x = 1, the expression simplifies to: 6 / 4 = 1.5

So, the derivative of (3sinx)/(2sinx+2cosx) is 1.5.