Question

The calculus of parametric equations: Let x = f(t) and y = g(t), where f and g are differentiable functions.

$\frac{dy}{dx}=.......$

Short answer

The value of $\frac{dy}{dx}=\frac{g\text{'}\left(t\right)}{f\text{'}\left(t\right)}$

Step-by-step solution

Step 1. Given information

$$\begin{gathered} x=f\left(t\right) \\ y=g\left(t\right) \end{gathered}$$

Step 2. Differentiate functions with respect to t.

$\frac{dx}{dt}=f\text{'}\left(t\right)$........(1)

$\frac{dy}{dt}=g\text{'}\left(t\right)$.............(2)

Step 3. Divide equation (2) by (1)

$$\begin{gathered} \frac{dy}{dt}÷\frac{dx}{dt}=\frac{g\text{'}\left(t\right)}{f\text{'}\left(t\right)} \\ \frac{dy}{dx}=\frac{g\text{'}\left(t\right)}{f\text{'}\left(t\right)} \end{gathered}$$