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

I get `m=-3`.

Explanation:

To use product rule instead of quotient, rewrite

`tan(xy)tan(x^2)=C`

Differentiate using the product and chain rules.

(I use `d/dx(uv) = u'v+uv'` for the product rule.)

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

At `(pi/3,pi/3)`, we have

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

`pisec^2(pi^2/9)tan(pi^2/9) + pi/3 sec^2(pi^2/9)tan(pi^2/9) dy/dx =0`

`dy/dx =(-pisec^2(pi^2/9)tan(pi^2/9))/(pi/3 sec^2(pi^2/9)tan(pi^2/9)) = - 3`