Assume we have a displacement field $u=u(x,z)=(u_x(x,z), u_z(x,z)))$ which takes the point $R=(x,z)$ to $r=R+u=(x+u_x(x,z), z+u_z(x,z))$. We want to find the tangent and normal to the curve which was $z=0$ before applying the displacement. This new curve is given by: $$r=(x+u_x(x,0),0+u_z(x,0))$$
This can be interpreted as a parameterized curve with parameter $x$ whose tangent vector is given by: $$e_t=r'(x)=(1+\frac{du_x(x,0)}{dx},\frac{du_z(x,0)}{dx})$$ Once the tangent is known, the normal is straightforward to find.
But the answer mentioned is instead:$$e_t=(1+\frac{du_x(x,0)}{dx},-\frac{du_z(x,0)}{dx})$$
Can someone point out the mistake?