How do you determine the instantaneous rate of change of y(x) = sqrt(3x + 1) for x = 1?

2 Answers
Mar 25, 2015

Apply the chain rule for derivatives and substitute x=1 in the result.

(d (y(x)))/(dx) = (1/2)(3x+1)^(-1/2)*(3)

(d (y(1)))/(dx) = 3/(2 sqrt(3+1)) = 3/4

(d (y(1)))/(dx) is the instantaneous rate of change of y(x) at x=1

Mar 25, 2015

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