Question

What is the slope of the tangent line of `tan(xy)-cot(x^2)= C`, where C is an arbitrary constant, at `(pi/3,pi/3)`?

Answer 1

`d/dx[tan(xy)-cot(x^2)=C]`

We will use implicit differentiation and the chain rule next.

`stackrel"product rule"overbrace(d/dx[xy])sec^2(xy)-d/dx[x^2]*-csc^2(x^2)=0`

`sec^2(xy)*(y+x[dy/dx])+2xcsc^2(x^2)=0`

We can now plug in `(pi/3,pi/3)` and solve for `dy/dx`, which will give us the slope of the tangent line at that point.

`sec^2(pi^2/9)*(pi/3+pi/3[dy/dx])+(2pi)/3csc^2((pi^2)/9)=0`

From here, it's just algebra.