Find dy/dx, if x=a² t² cotθ and y=at sin⁡θ

The correct answer is tanθsinθ/2at.

To find the derivative
dydx
when x and y are given in terms of a parameter t (with a and θ treated as constants), we use the parametric differentiation formula:
dydx=dydtdxdt

Step 1: Differentiate x with respect to t
Given:
x=a2t2cotθ
Differentiating with respect to t:
dxdt=dtdta2t2cotθ
Since a2 and cotθ are constants:
dxdt=a2cotθ·ddtt2=a2cotθ·2t=2a2tcotθ

Step 2: Differentiate y with respect to t
Given:
y=atsinθ
Differentiating with respect to t:
dydt=ddtatsinθ
Since a and sinθ are constants:
dydt=asinθ·ddt(t)=asinθ

Step 3: Compute dydx
Now substitute our calculated derivatives into the parametric formula:
dydx=asinθ2a2tcotθ
Simplifying the fraction by dividing the numerator and denominator by a gives:
dydx=sinθ2atcotθ

Step 4: Express in terms of standard functions
Since
1cotθ=tanθ
we can rewrite the expression as:
dydx=tanθsinθ2at