Second derivatives of parametric curves

Understand the idea

The second derivative $\displaystyle d^2y/dx^2$ measures how the Cartesian slope changes with the horizontal coordinate. After finding $\displaystyle dy/dx$ as a function of the parameter, differentiate that slope with respect to the parameter and divide by $\displaystyle dx/dt$ once more.

This is a chain-rule calculation: differentiating a slope expression with respect to $\displaystyle t$ does not yet give its rate of change with respect to $\displaystyle x$ . The extra division converts between the two input scales. A curve can have changing parameter speed without changing its shape.

Where the formula applies, a positive second derivative indicates concave-up behavior as a graph of $\displaystyle y$ against $\displaystyle x$ , and a negative value indicates concave-down behavior. Interpret this separately from whether the object is moving left or right along the curve.

\frac{d^2y}{dx^2}=\frac{\dfrac{d}{dt}\left(\dfrac{y'(t)}{x'(t)}\right)}{x'(t)}

A method you can apply

Compute $\displaystyle dy/dx$ and simplify it as a function of $\displaystyle t$ .
Differentiate the simplified slope with respect to $\displaystyle t$ .
Divide by $\displaystyle x'(t)$ again and evaluate only where the required derivatives are defined.

Worked example

For $\displaystyle x=t^2$ and $\displaystyle y=t^3$ , find $\displaystyle d^2y/dx^2$ at $\displaystyle t=1$ .

For $\displaystyle t\ne0$ , the first derivative is $\displaystyle 3t^2/(2t)=3t/2$ .
Its derivative with respect to $\displaystyle t$ is $\displaystyle 3/2$ .
Dividing by $\displaystyle x'=2t$ gives $\displaystyle d^2y/dx^2=3/(4t)$ .

At

\displaystyle t=1

, the second derivative is

\displaystyle 3/4

, so the curve is concave up there.

Try it yourself

For $\displaystyle x=t^3$ and $\displaystyle y=t^2$ , find $\displaystyle d^2y/dx^2$ at $\displaystyle t=1$ .

Show solution and reasoning

First $\displaystyle dy/dx=2/(3t)$ . Differentiating with respect to $\displaystyle t$ gives $\displaystyle -2/(3t^2)$ , then dividing by $\displaystyle 3t^2$ gives $\displaystyle -2/(9t^4)$ . At $\displaystyle t=1$ , the answer is $\displaystyle -2/9$ .