If `x=at^4`, `y=bt^3`, then find `(dy)/(dx)`.

Explanation

This is called implicit differentiation or indirect differentiation. Here, both the given terms are differentiated independently w.r.t. a third variable and then combined together. Differentiating x and y w.r.t t, we get
`(dx)/(dt)=4at^3`, `(dy)/(dt)=3bt^2`
Dividing `(dy)/(dt)` by `(dx)/(dt)`, we get
`(dy)/(dx)=(dy)/(dt)xx(dt)/(dx)=(3bt^2)/(4at^3)=(3b)/(4at)`