I need to find the directional derivative of $x+y-z=\ln z$ in the point $(1,0,1)$ in the direction which is formed at the angle of $\arctan({4 \over 3})$ with the positive values of $x$ axis.
The gradient $\nabla f=\langle f_x, f_y,f_z\rangle=\langle z_x, z_y,z_z\rangle$.
I got the partial derivative for $z_x=z_y=\frac{z}{z+1}$ but I don't understand how to differentiate the given function implicitly for $z$.
Any suggestions would be highly welcome.