Equations in terms of x and y can be defined two different ways. One is explicitly, y = f (x), and the other, implicitly.

EXPLICITLY:

y = x ² + 3x - 4

IMPLICITLY:

x ² + y ² = 4

We know how to take the derivative of the first.

If we want to find the derivative of the implicitly defined function, I would first have to solve for y in terms of x. Then the question would be this: which function would I want to use? It would depend on what we were asked to find. As you should know, calculus is not a guessing game, so there has to be an easier way to find the derivative. The easy way to find this derivative is to find it by implicit differentiation.

When performing implicit differentiation, you must remember that y is a function of x. Hence, every time you take the derivative of y, you are performing a chain rule. Therefore, you must tack on a dy/dx. After taking the derivatives, solve for dy/dx in terms of x and y.

EXAMPLE 1:

SOLUTION:

EXAMPLE 2:

EXAMPLE 3:

EXAMPLE 4:

EXAMPLE 5:

2.

Now implicitly differentiate dy/dx.

EXAMPLE 6:

You could simplify the above result, but it is fairly complicated. Also notice that finding the second derivative is not the easiest thing to do. If you take your time, you will be able to find it.

We know how to find the tangent line to the curve, but another type of line you might be asked to find is the normal line. So what is the normal line?

DEFINITION:

A line is normal to a curve at a point if it is perpendicular to the curve's tangent there. This line is called the normal to the curve at the point.

EXAMPLE 7:

Verify that (-1, 3) is a point on the curve x ² y ² = 9. Then find the tangent and normal line to the curve at that point.

SOLUTION: How do you verify that (-1, 3) is a point on the curve x ² y ² = 9? All you have to do is plug the point into the equation and determine if it equals 9.

Now, I need to find the first derivative, which will be the formula for the slope of the tangent line.

Now to find the slope and the equation of the tangent line at the point (-1, 3).

Now to find the slope and the equation of the normal line. Remember that the perpendicular slope is the negative reciprocal of the tangent line's slope.

EXAMPLE 8:

Verify that (0, p ) is a point on the curve x ² cos ² y - sin y = 0. Then find the tangent and normal lines to the curve at that point.

SOLUTION: First, let us verify that the point is on the curve.

Now to find the first derivative.

Now to find the slope and the equation of the tangent line.

The tangent line is a horizontal line, so the normal line will be a vertical line. The slope of the tangent line was 0, so the slope of the normal line is undefined. Therefore, the equation for the normal line is x = 0.