Implicit Differentiation #6

Question

Find the slope of the tangent line to the curve

at the point .

Answer 1

5sin(x) + 4cos(y) - 4sin(x)cos(y) + x = 4π

5cos(x) - 4sin(y)dy/dx - 4(-sin(x)sin(y)dy/dx + cos(x)cos(y)) + 1 = 0

5cos(x) - 4sin(y)dy/dx + 4sin(x)sin(y)dy/dx - 4cos(x)cos(y) + 1 = 0

5cos(x) - 4cos(x)cos(y) + 1 = 4sin(y)dy/dx - 4sin(x)sin(y)dy/dx

5cos(x) - 4cos(x)cos(y) + 1 = 4sin(y)dy/dx - 4sin(x)sin(y)dy/dx

4dy/dx(sin(y) - sin(x)sin(y)) = 5cos(x) - 4cos(x)cos(y) + 1

dy/dx = (5cos(x) - 4cos(x)cos(y) + 1)/(4 (sin(y) - sin(x)sin(y)))

dy/dx = (5cos(4π) - 4cos(4π)cos(7π/2) + 1)/(4(sin(7π/2) - sin(4π)sin(7π/2)))

dy/dx = (5*1 - 4*1*0 + 1)/(4(-1 - 0*-1)))

dy/dx = -6/4

dy/dx = -3/2