Question

If $x=a{t}^4,y=b{t}^3$, then find $\frac{dy}{dx}$.

Answer 1

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
$\frac{dx}{dt}=4a{t}^3,\frac{dy}{dt}=3b{t}^2$
Dividing $\frac{dy}{dt}by\frac{dx}{dt}$, we get
$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}=\frac{3b{t}^2}{4a{t}^3}=\frac{3b}{4at}$.