How to find the correct representation for the slope of the tangent line to the curve cosxy + 3sin^2 x + 2y.

Answer 1

Take the derivative with respect to x:

d(dx(cos(xy) + 3sin²⁽x) + 2y) =

First term:

d/dx(cos(xy)) = -sin(xy) * d/dx(xy)

Use product rule to evaluate second part::

d/dx(xy) = x(dy/dx) + y

Therefore:

d/dx(cos(xy)) = -sin(xy)(x(dy/dx) + y )

Second term:

d/dx(3sin²(x)) = 6sin(x)cos(x)

Third term:

d/dx(2y) = 2dy/dx

So:

d(dx(cos(xy) + 3sin²⁽x) + 2y) = -sin(xy)(x(dy/dx) + y ) + 6sin(x)cos(x) +2dy/dx