Creative partial derivative example

Question

let $f \epsilon C^1$ with $$f(1,1)=1\\ f_1(1,1)=a\\ f_2(1,1)=b$$

And now let $$h(x)=f(x,f(x,x))$$ find $h'(1)$

My approach :

$$\frac{dh}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial (f(x,x))} \frac{\partial f}{\partial x}$$

Since $$(*) \frac{\partial f(x,x)}{\partial x}= \frac{2\partial f}{\partial x}$$ I believe (*) is this case since $f$ is eventually reduced to a one variable. But This doesn't seem to be the answer,

Answer 1

Just a Hint

Write $h(x)=f(x,y)$ and $y=f(x,x)$.

then

$$h'(x)=f'_x+f'_y.y'_x$$

$$=f'_x+f'_y(f'_x+f'_y)$$