Question

If x and y are connected parametrically by the equation, without eliminating the parameter, find dy/dx x=2at^2 ,y=at^4

Answer 1

it is given that x = 2at² and y = at⁴
step 1 :- differentiate x with respect to t,
$\frac{dx}{dt}=\frac{d}{dt}(2at^2)\\\\=4at^{2-1}=4at$

step 2 :- differentiate y with respect to t,
$\frac{dy}{dt}=\frac{d}{dt}(at^4)\\\\=4at^{4-1}=4at^3$

now, dividing dy/dt with dx/dt
e.g., $\frac{dy/dt}{dx/dt}=\frac{4at^3}{4at}$

$\frac{dy}{dx}=t^2$

hence, the value of dy/dx is t²