In this explainer, we will learn how to find second derivatives and higher-order derivatives of parametric equations by applying the chain rule.

Parametric equations are a way in which we can express the variables in an equation in terms of another parameter. For example, if we have an equation in terms of the variables and , then we could write parametric equations for these variables in terms of a parameter, , as follows:

### Note

Parametric equations can be used in conjunction with any coordinate system, not only Cartesian. For example, if we wanted to parameterize some polar coordinates, we would express and in terms of a parameter.

We can find the derivative of with respect to in terms of the parametric equations using the following definition.

### Definition: Derivative of a Parametric Equation

Let and be differentiable functions, such that and are a pair of parametric equations:

Then we can define the derivative of with respect to as when .

The first derivative of an equation can be a very useful tool for finding equations of tangents and normals to the curve or calculating gradients along the curve. The second derivative, or , can also tell us useful information about the concavity of the curve.

You may think that we can find the second derivative by finding the second derivatives of the parametric equations with respect to and dividing by , similar to how we did for the first derivative. However, this does not work as you will see below.

We find the second derivative of with respect to by differentiating the first derivative with respect to :

Finding this second derivative in terms of the parametric equations is not simple, since the equation we have for the first derivative is in terms of our parameter, . In order to perform this differentiation with respect to , we will need to use the chain rule. Recall the definition of the chain rule.

### Definition: The Chain Rule

Given a function that is differentiable at and a function that is differentiable at , their composition which is defined by is differentiable at and its derivative is given by

We know how to find in terms of , since we can find the first derivative of a parametric equation. However, we need to differentiate this with respect to to find the second derivative. In order to do this, we will need to use a different form of the chain rule, which is as follows:

Applying this to our equation for the second derivative, we get

Now, we have , which is in terms of ; however, is in terms of . Here, we can use the inverse function theorem, which tells us that, for derivatives which are nonzero,

Substituting this into our equation gives us the following formula for finding the second derivative of parametric equations:

### Definition: Second Derivative of a Parametric Equation

Let and be differentiable functions such that and are a pair of parametric equations: Then, we can define the second derivative of with respect to as when .

Let us now look at an example of how we can find the second derivative of a parametric equation.

### Example 1: Finding the Second Derivative of Parametric Equations

Given that and , find .

### Answer

The first step in finding the second derivative of these parametric equations is to find the first derivative. We can do this by using the formula

First, we can differentiate with respect to . Since is a polynomial in terms of , we use polynomial differentiation. First, multiply each term by its power of , and then reduce the power of by one. This gives us

Similarly, we also need to differentiate with respect to . This gives us

Substituting these back into our formula for the first derivative, we obtain

We are now ready to use the formula for the second derivative, which is as follows:

We have already found , so all we need to do is find the derivative of with respect to . By differentiating, we obtain

We have now found all the parts for our equation for the second derivative; we can substitute them in, which will give us our solution of

We can evaluate the second derivative of a parametric equation at a given point as we can see in the next example.

### Example 2: Evaluating the Second Derivative of Parametric Equations at a Given Point

If and , find at .

### Answer

Since we are trying to find the second derivative of with respect to , where and are in parametric form, we can use the following equation:

We can start by differentiating and with respect to . Using polynomial differentiation, we get and

Using this, we can find

Next, we need to differentiate with respect to . In doing this, we obtain

We are now ready to find . Substituting what we have found into the formula, we obtain

Now all that we need to do is evaluate our second derivative at the given point, . When we do this, we reach our solution of

The second derivative of an equation can also tell us about the concavity of the function at that point.

### Definition: The Concavity of a Function at a Point

For a function, , which is twice differentiable and exists at some ,

- if , then is concave upward at ;
- if , then is concave downward at ;
- if , then may be concave upward or
downward, or there may be a point of inflection at .

We will need to do another test to check which is the case.

Let us now look at an example of how we can find the concavity of a parametric curve at a given point.

### Example 3: Determining the Concavity of a Parametric Curve at a Given Point

Consider the parametric curve and . Determine whether this curve is concave up, down, or neither at .

### Answer

We have been asked about the concavity of a curve, so we will need to evaluate the second derivative of the curve at the given point. Since this is a parametric curve, we can use the following formula to find its second derivative:

Let us start by finding and . We can do this by using trigonometric differentiation. We have that and

So we find that

Now we can differentiate with respect to . Using the differentials of reciprocal trigonometric functions, we have that

Now we are able to substitute these into the formula for the second derivative. In doing this, we obtain

Now that we have found the second derivative of the curve, we need to substitute in the value of the parameter at the point at which we are trying to find the concavity. The given point is , so we substitute this into our equation which gives us

We can see that the second derivative at is negative, so we have that

Therefore, our function must be concave downward at this point.

In our final example, let us see how we can find a function which involves the second derivative of parametric equations.

### Example 4: Finding the Second Derivative of a Function Defined by Parametric Equations

If and , find .

### Answer

In order to solve this question, we first need to find . We can use the formula

In order to find , we need to find and . We can expand the binomial terms in and to obtain

We differentiate these with respect to to get

Using what we have just found, we can say that

In order to find the second derivative, we need to differentiate with respect to . In order to do this, we will need to use the quotient rule for differentiation. The quotient rule tells us that if we have a function of the form , then we find its derivative using

In our case, and . Therefore, and . Substituting these values into our formula, we obtain

We can simplify the numerator to give us

Now, we are ready to find . Substituting into the formula, we have

In order to find the solution to the problem, all we need to do is multiply by . In doing this, we reach our solution of

We have now seen how to find the second derivative of parametric equations and how we can use it to find the concavity of a parametric curve. Let us recap some key points.

### Key Points

- We can find the second derivative of parametric equations using the formula where when .
- We can use the second derivative to find the concavity of the curve at different points.