How to find the derivative of g(x) = sec^−1 (x^3).

Answer 1

y = sec^-1(x³)

sec(y) = x³

Take implicit derivative both sides

sec(y)tan(y) dy/dx = 3x²

dy/dx = 3x²cos(y)ctn(y)

Draw triangle for sec(y) = x³/1

cos(y) = adj/hyp=1/x³

ctn(y) = adj/opp = 1/√(x⁶-1)

dy/dx = 3x²(1/x³)/√(1-x⁶) = 3/(x√(x⁶-1)) = 3/(x⁴√(1-1/x⁶))