The graph of is continuous for all real numbers except for one value, x = c. Use the rate of change of the equation to help you find c and classify the discontinuity you find in the derivative.

Difficulty:

Solution

To calculate the rate of change, you can do one of two things: Either differentiate implicitly or solve for y and then derive according to the Quotient Rule. I tried to steer you away from that with my leading problem title, because if you solve for y, the point of discontinuity becomes immediately obvious. So, differentiate implicitly (i.e. with respect to x) to humor me, remembering that the derivative of 4xy will require the Product Rule (since it contains two variables). To make it clear what I'm doing, I will set up the derivative before I actually start, denoting where I plan on deriving with a prime ( ' ) symbol.

Remember, the derivative of 3 (with respect to x) is 0. Any constant's derivative is 0. Now solve for dy/dx to get the rate of change:

The derivative will be undefined whenever the denominator is equal to 0, so in other words when x = 1/4. This suggests that the derivative is infinitely large when x = 0, indicating that the tangent line would be vertical at that point. That means that the discontinuity of the derivative is nonremovable (i.e. cannot be repaired with a finite number of redefined points). If you graph the function, this is confirmed; the equation has a vertical asymptote there.