What Parametric Slope Means
When a curve is defined parametrically, you are given separate equations for and in terms of a third variable, usually . Instead of describing the curve directly as , the parameter tells you where you are on the curve as it changes.
The important idea is that the slope is still rise over run. The only difference is that both the vertical change and the horizontal change are expressed in terms of . So instead of differentiating directly with respect to , we calculate how quickly and are changing with respect to .
The Formula
Suppose a curve is defined parametrically by
Differentiate both equations with respect to :
Then use the fact that the slope of the curve is the rate of change of with respect to . Divide the two derivatives:
This is the central formula to memorize for parametrically defined curves:
In words: differentiate with respect to the parameter, differentiate with respect to the parameter, and divide.
A Step-by-Step Method
- Differentiate with respect to . This gives .
- Differentiate with respect to . This gives .
- Divide: .
- Substitute the given parameter value, if the problem asks for the slope at a particular point or parameter value.
- Simplify your answer and check whether the denominator is zero. A zero denominator means the tangent is vertical rather than giving an ordinary finite slope.
Worked Examples
Suppose
Find the slope of the curve at .
Suppose
Find at .
Consider the parametric equations
Find the slope of the curve at the point corresponding to .
Suppose
Find the slope when .
Horizontal Tangents
A horizontal tangent has slope zero. For a parametrically defined curve,
A fraction is zero when its numerator is zero, as long as the denominator is not also zero. Therefore, to find horizontal tangents:
- Set equal to zero.
- Solve for the parameter value or values.
- Check that is not zero at those values.
- Find the corresponding point if the problem asks for coordinates.
Let
Find the parameter value where the curve has a horizontal tangent.
Vertical Tangents
A vertical tangent occurs when the horizontal rate of change is zero while the vertical rate of change is not zero. In parametric form, that means
So to find vertical tangents:
- Set equal to zero.
- Solve for the parameter value or values.
- Check that is not zero.
- Find the corresponding point if required.
Suppose
Find the parameter value where the curve has a vertical tangent.
Why the Formula Works
The formula for parametric slope is not a new rule that appears out of nowhere. It comes directly from the chain rule.
Since both and depend on , think of as a function of and as a function of .
Using the chain rule,
Now solve this equation for :
So the familiar parametric derivative formula is really a direct consequence of the chain rule. This is also why it is important to understand what the parameter represents: it is the variable through which both coordinates are changing.
Common Mistakes
The slope is vertical change divided by horizontal change. Therefore the correct formula is , not . Think "dy over dx," so the derivative of goes on top and the derivative of goes on the bottom.
If the equations are written in terms of , you should calculate and . Do not differentiate one equation with respect to and the other with respect to .
If a problem asks for the slope at , substitute the parameter value into your final expression for . Do not automatically substitute the x-coordinate unless the problem specifically gives you a way to determine the corresponding parameter value.
If is zero, the slope formula has a zero denominator. That does not mean "the slope is zero." It means you should check whether is nonzero. If it is, the tangent is vertical.
Practice Problems
The curve is defined by . Find at .
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The curve is defined by . Find the slope at .
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For , find the parameter value where the curve has a horizontal tangent.
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For , find the parameter value where the curve has a vertical tangent.
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