If you need to do this from the definition, the details will depend on your textbook and instructor's choice of definition:
You'll have one of:
lim_(xrarr1)(y(x)-y(1))/(x-1) or lim_(hrarr0)(y(1+h)-y(1))/h
Definition 1:
lim_(xrarr1)(y(x)-y(1))/(x-1)=lim_(xrarr1)(sqrt(3x+1)-sqrt(3(1)+1))/(x-1)
=lim_(xrarr1)(sqrt(3x+1)-2)/(x-1)
Notice that substitution leads to indeterminate form: 0/0. The trick (technique) to try in this case is to rationalize the numerator. Multiply numerator and denominator by the conjugate of the numerator. (Multiply the top and bottom by (sqrt(3x+1) + 2)
lim_(xrarr1)(y(x)-y(1))/(x-1)=lim_(xrarr1)[((sqrt(3x+1)-2))/(x-1) ((sqrt(3x+1) + 2))/((sqrt(3x+1) + 2))]
=lim_(xrarr1)(sqrt(3x+1)^2-2^2)/((x-1)(sqrt(3x+1) + 2)
=lim_(xrarr1)(3x+1-4)/((x-1)(sqrt(3x+1) + 2)
=lim_(xrarr1)(3x-3)/((x-1)(sqrt(3x+1) + 2)
=lim_(xrarr1)(3(x-1))/((x-1)(sqrt(3x+1) + 2) (Still form 0/0)
=lim_(xrarr1)3/(sqrt(3x+1) + 2) (No longer form 0/0)
=3/(sqrt(3(1)+1) + 2)=3/(2+2)=3/4
Definition 2:
lim_(hrarr0)(y(1+h)-y(1))/h =lim_(hrarr0)(sqrt(3(1+h)+1)-sqrt(3(1)+1))/h
=lim_(hrarr0)(sqrt(3h+4)-2)/h
Notice that substitution leads to indeterminate form: 0/0. The trick (technique) to try in this case is to rationalize the numerator. Multiply numerator and denominator by the conjugate of the numerator. (Multiply the top and bottom by (sqrt(3h+4) + 2)
lim_(hrarr0)(y(1+h)-y(1))/h =lim_(hrarr0)((sqrt(3h+4)-2))/h ((sqrt(3h+4) + 2))/((sqrt(3h+4) + 2))
=lim_(hrarr0)((sqrt(3h+4)^2-2^2))/(h(sqrt(3h+4) + 2))
=lim_(hrarr0)(3h+4-4)/(h(sqrt(3h+4) + 2)) , (still form 0/0)
=lim_(hrarr0)(3h)/(h(sqrt(h+4) + 2)) , (still form 0/0)
=lim_(hrarr0)3/(sqrt(h+4) + 2) , (no longer form 0/0)
=3/(sqrt(0+4) + 2)=3/(2+2)=3/4