(* define position vector as a function of x *)
r[x_,y_,z_]:={x,y,z};
(* define magnetic moment μ *)
m:={0,0,μ};
(* test *)
m.r[x,y,z]
(* physical constant *)
pars={μ0-> 4*π*1*^-7,μ->8.8};

(* define dipole field as a function of r vector *)
B[r_]:=(((μ0)/(4*π)))(((3r(m.r))/((Norm[r])^5))-(m/((Norm[r])^3)))

(* calculate the dBz/dx assuming one centimeter offset in z *)
(∂_x)B[r[x,y,z]]/.y->0/.z->0.1

(* define derivative using assumption to get rid of derivatives of absolute value *)
dBdx=FullSimplify[(∂_x)B[r[x,y,z]]/.y->0/.z->0.1,Assumptions->{Element[x,Reals]}]

(* estimate typical values of dBdx *)
dBdx/.pars/.x->1*^-18

(* plot results *)
Plot[
dBdx[[3]]/.pars,
{x,-1*^-18,1*^-18},
PlotRange->{-2*^-18,2*^-18}
]

(* define derivative using assumption to get rid of derivatives of absolute value *)
dBdz=FullSimplify[(∂_z)B[r[x,y,z]]/.x->0/.y->0,Assumptions->{Element[z,Reals]}]

(* estimate typical values of dBdz *)
dBdz/.pars/.z->0.01