Question

If x = e^θ, (sin θ – cos θ), y = e^θ (sin θ + cos θ) then `dy/dx` at θ = `π/4` is ______.

Options

• 1

• 0

• `1/sqrt(2)`

• `sqrt(2)`

Answer 1

If x = e^θ, (sin θ – cos θ), y = e^θ (sin θ + cos θ) then `dy/dx` at θ = `π/4` is 1.

Explanation:

x = e^θ (sin θ – cos θ) and y = e^θ (sin θ + cos θ)

∴ `dx/(dθ)` = e^θ (cos θ + sin θ) + (sin θ – cos θ) e^θ

= e^θ [2 sin θ]

and `dy/dx` = e^θ (cos θ – sin θ) + (sin θ + cos θ) e^θ

= e^θ [2 cos θ]

∴ `dy/dx = (dy//dθ)/(dx//dθ) = (e^θ[2 cos θ])/(e^θ[2 cos θ])`

`\implies dy/dx` = cot θ

`\implies dy/dx |_(θ = π/4) = cot  π/4` = 1