Differentiate with respect to x: w=4x^2 + 3sin(2x)

We will split this up and differentiate each part separately.

We can differentiate 4x² using our normal rules for differentiating; we multiply the coefficient by the power and then subtract one from the power. This gives us; 4x2=8, x^2-1=x¹ => 8x¹ which can be written as 8x.

To differentiate 3sin(2x) we have to use the product rule. The product rule states that if y=uv, then : dy/dx= u dv/dx + v du/dx. In this case our y=3sin(2x), so we have u=3 and v=sin(2x), we differentiate and find that du/dx=0, because 3 is a constant, and we find that dv/du=2cos(2x). Now using the product rule we find dy/dx=3x[2cos(2x)]+[2cos(2x)]x0=6cos(2x)+0 = 6cos(2x). 

Bringing both parts together we get  dw/dx=8x + 6cos(2x)