Theory question 109
Differentiation(a) Differentiate (x2 + 1)/(x + 1)2 with respect to x.
(b)(i) Evaluate the determinant (1 2 −1 / 2 3 −1 / −1 1 3) (rows separated by /).
(ii) Using the answer in 9(b)(i), solve the system of equations: x + 2y − z = 4; 2x + 3y − z = 2; −x + y + 3z = −1.
Show the model answer
Model answer
(a) Quotient rule: [2x(x + 1)² − 2(x + 1)(x² + 1)]/(x + 1)⁴ = [2x(x + 1) − 2(x² + 1)]/(x + 1)³ = 2(x − 1)/(x + 1)³.
(b)(i) Δ = 1(9 + 1) − 2(6 − 1) − 1(2 + 3) = −5.
(ii) Cramer's rule: Δₓ = 25, Δᵧ = −15, Δ_z = 15.
x = 25/−5 = −5, y = −15/−5 = 3, z = 15/−5 = −3.
Explanation
Check the solution in all three equations.