Finding the tangent line using implicit differentiation

Question

My professor wrote this problem on the board as a challenge:

Find the tangent line at (0,0) to the curve defined implicitly below. $$\ln(1+x+y)=\left( x^{42} e^y + \cos(xy)\sin(xy)\right)^{2015} \left( ye^{x\cos y} + 429 \sin(y \cos(x))\right)^{257} + 2x$$

I tried differentiating the equation, but I gave up because it was just too messy. Is there an easier way?

Answer 1

Write $$\ln(1+x+y)=f(x,y)^{2015}g(x,y)^{257}+2x$$

Now differentiate:

$$\frac{1+y'}{1+x+y}=2015y'f'(x,y)f(x,y)^{2014}g(x,y)^{257}+257y'g'(x,y)g(x,y)^{256}f(x,y)^{2015}+2$$

Since $f(0,0)=g(0,0)=0$, $$1+y'=2$$