### Parametric Differentiation

#### Definition

Suppose a 2D curve is defined parametrically, where $x=x(t)$ and $y=y(t)$. Then, provided that $\frac{\mathrm{d} x}{\mathrm{d} t}\neq0$, the derivative of $y$ with respect to $x$ is given by:

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{^{\Large{\mathrm{d} y} }/_{\Large{\mathrm{d} t} } }{^{\Large{\mathrm{d} x} }/_{\Large{\mathrm{d} t} } }.$

This result is derived by using the chain rule, $\dfrac{\mathrm{d} y}{\mathrm{d} t}=\dfrac{\mathrm{d} y}{\mathrm{d} x}\dfrac{\mathrm{d} x}{\mathrm{d} t},$ and dividing through by $\dfrac{\mathrm{d} x}{\mathrm{d} t}$.

#### Worked Example

###### Example 1

Consider the parametrically defined function $x(t)=t^2-\cos{t};\: y(t)=t+\mathrm{e}^{\large{2t} }$. Find $\dfrac{\mathrm{d} y}{\mathrm{d} x}$, and find the point where $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ is not defined.

###### Solution

Recall that $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\dfrac{^{\Large{\mathrm{d} y} }/_{\Large{\mathrm{d} t} } }{^{\Large{\mathrm{d} x} }/_{\Large{\mathrm{d} t} } }.$

The derivative of $y$ with respect to $t$ (found using standard derivatives) is:

\begin{align} \dfrac{\mathrm{d} y}{\mathrm{d} t} &= \dfrac{\mathrm{d} }{\mathrm{d} t}\bigl[t+\mathrm{e}^{\large{2t} }\bigl] \\ &= 1+2\mathrm{e}^{\large{2t} }. \end{align}

The derivative of $x$ with respect to $t$ is:

\begin{align} \dfrac{\mathrm{d} x}{\mathrm{d} t} &= \dfrac{\mathrm{d} }{\mathrm{d} t}\bigl[t^2-\cos{t}\bigl] \\ &= 2t-(-\sin{t}) \\ &= 2t+\sin{t}. \end{align}

The derivative $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ is therefore:

\begin{align} \dfrac{\mathrm{d} y}{\mathrm{d} x} &= \dfrac{^{\Large{\mathrm{d} y} }/_{\Large{\mathrm{d} t} } }{^{\Large{\mathrm{d} x} }/_{\Large{\mathrm{d} t} } } \\ &= \dfrac{1+2\mathrm{e}^{\large{2t} } }{2t+\sin{t} }. \end{align}

The point where $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ is not defined is the point where $\dfrac{\mathrm{d} x}{\mathrm{d} t}=0$, since this would give $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\dfrac{^{\Large{\mathrm{d} y} }/_{\Large{\mathrm{d} t} } }{0}$, which is clearly undefined.

Here $\dfrac{\mathrm{d} x}{\mathrm{d} t}=2t+\sin{t}$, so $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ is undefined where $2t+\sin{t}=0,$. This equation holds at $t=0$ (since $\sin{0}=0)$.

The value of $x$ where $t=0$ is:

$x(0)=0^2-\cos{0}=-1.$

The value of $y$ where $t=0$ is:

$y(0)=0+\mathrm{e}^0=1.$

Hence $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ is undefined at the point $(-1,1)$.

The following graph shows the given parametric curve and the line tangent to the curve at the point $(-1,1)$, where the gradient is undefined.

#### Video Examples

###### Example 1

Prof. Robin Johnson finds $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ given a parametrically defined function $x(t)=2t-\cos{t}$; $y(t)=t^2+\sin{t}$.

###### Example 2

Prof. Robin Johnson finds $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ and $\dfrac{\mathrm{d} ^2y}{\mathrm{d} x^2}$ given a parametrically defined function $x(t)=2t^2+t$; $y(t)=t^3-t$.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

#### More Support

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