# Existence of derivative

**Right hand and left hand derivatives**

- The right-hand derivative of
*f*at*x*=*a*denoted by*f*â€²(*a*^{+}) is defined by*f*â€²(*a*^{+}) =*h*â†’ 0, the point*B*moving along the curve tends to*A,*i.e.,*B*â†’*A*then the chord*AB*approaches the tangent line*AT*at the point*A*and then*Ï†*â†’*Ïˆ**f*â€²(*a*^{+}) = = tan*Ïˆ* - The left-hand derivative of
*f*at*x*=*a*denoted by*f*â€²(*a*^{â€“}) is defined by*f*â€²(*a*^{â€“}) =*h*â†’ 0, the point*C*moving along the curve tends to*A,*i.e.,*C*â†’*A*then the chord*CA*approaches the tangent line*AT*at the point*A*then*f*â€²(*a*^{â€“}) = - At
*A*,*f*(*x*) is differentiable if*f*â€²(*a*^{+}) and*f*â€²(*a*^{â€“}), both exist and are finite. In other words*f*(*x*) is differentiable at*x*=*a*, if unique tangent can be drawn at this point.