Given
e ˣʸ = sec(x ²)
take the derivative of both sides:
d/dx [e ˣʸ] = d/dx [sec(x ²)]
Use the chain rule:
e ˣʸ d/dx [xy] = sec(x ²) tan(x ²) d/dx [x ²]
Use the product rule on the left, and the power rule on the right:
e ˣʸ (x dy/dx + y) = sec(x ²) tan(x ²) (2x)
Solve for dy/dx :
e ˣʸ (x dy/dx + y) = 2x sec(x ²) tan(x ²)
x dy/dx + y = 2x e ⁻ˣʸ sec(x ²) tan(x ²)
x dy/dx = 2x e ⁻ˣʸ sec(x ²) tan(x ²) - y
dy/dx = 2e ⁻ˣʸ sec(x ²) tan(x ²) - y/x
Since e ˣʸ = sec(x ²), we simplify further to get
dy/dx = 2 tan(x ²) - y/x