Differentiating Parametric Equations

MathsDirect

It is not always easy to write equations in the form

It is sometimes easier to write the relationship between x & y in terms of a third variable, called a parameter. This parameter is usually represented by the letter t.

For example	;
So , when t = 3	;

To find the gradient of a curve given parametrically, you have two options.

1 Rewrite the equation in cartesian form.

2 Differentiate parametrically, using the chain rule.

The problem with the first method is that equations are generally given parametrically, specifically because they are hard to write in a cartesian form.

The example above, however is easy to convert, as an example.

Make t the subject of the x equation.
Substitute into the y equation
Differentiate as usual

In general, however, you should not use this method.

Parametric Differentiation

To differentiate equations given parametrically, you should use the chain rule,

That is, you differentiate y and x separately, with respect to the parameter t. This can be rewritten

In the example above

		This is the same answer that we got before.

Another example

Find the gradient of the curve described by the parametric equations

This is an ellipse

In this case it would be difficult to write in a cartesian form. Differentiating separately gives.



Combining these gives

On the next page there is an example of finding the equation of a tangent to a curve given parametrically.

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