Find the value of dydx at θ=π4 if x=aeθ(sinθ−cosθ) and y=aeθ(sinθ+cosθ).

y=aeθ(sinθ+cosθ)
x=aeθ(sinθ−cosθ)
Applying parametric differentiation,
dy/dx = (dy/dθ)/(dx/dθ) —(1)
Now,
dy/dθ = aeθ(cosθ−sinθ) + aeθ(sinθ+cosθ)
Applying product rule: we get
= 2aeθ(cosθ)
dx/dθ = aeθ(cosθ+sinθ) + aeθ(sinθ−cosθ)
= 2aeθ(sinθ)
Substituting the values of dy/dθ and dx/dθ in (1),
dy/dx = 2aeθcosθ / 2aeθsinθ = cotθ
Now, dy/dx at θ=π/4 [cotθ]θ=π/4 = cot(π/4) = 1.