The derivative f'(x) = d/dx(x^4 + 2x^3 - x + 1) = 4x^3 + 6x^2 - 1.

The derivative f'(x) = 4x^3 - 12x^2 + 12x - 24.

The derivative is f'(x) = 4x^3 - 12x^2 + 12x - 4.

Using the power rule, f'(x) = 4x^3 - 12x.

f'(x) = 4x^3.

The derivative f'(x) = d/dx(x^5 + 2x^3 - x) = 5x^4 + 6x^2 - 1.

The derivative f'(x) = d/dx(x^5) = 5x^4.

The left-hand derivative is -1 and the right-hand derivative is 1, hence the derivative at x = 0 is undefined.

f'(x) = 1/(2√x).

The derivative of sin(x) is cos(x).

The derivative of sin^(-1)(x) is 1/√(1-x^2)

The derivative f'(x) = d/dx(3x^2 + 5x - 7) = 6x + 5.

The derivative of x^3 is 3x^2.

The derivative of y = sin^(-1)(x) is 1/√(1-x^2)

The derivative of \( y = \tan^{-1}(x) \) is \( \frac{1}{1+x^2} \).

The determinant of a 1x1 matrix is simply the value of the single element. Therefore, the determinant of [[5]] is 5.

The determinant of a 2x2 matrix is calculated as ad - bc.

The determinant of a 2x2 matrix is calculated as ad - bc.

Determinant of F = (2*3) - (0*0) = 6.

Det(G) = 1(2*6 - 3*3) - 1(1*6 - 1*3) + 1(1*3 - 1*2) = 1(12 - 9) - 1(6 - 3) + 1(3 - 2) = 3 - 3 + 1 = 1.

Det(G) = 1(5*9 - 6*8) - 2(4*9 - 6*7) + 3(4*8 - 5*7) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = -3 + 12 - 9 = 0.

Determinant of G = (1*4) - (2*2) = 4 - 4 = 0.

Det(G) = (2*2) - (1*1) = 4 - 1 = 3.

Det(G) = (2*2) - (2*2) = 4 - 4 = 0.

The determinant of G is (3*4) - (2*1) = 12 - 2 = 10.

Det(G) = (5*3) - (4*2) = 15 - 8 = 7.

Det(G) = (5*8) - (6*7) = 40 - 42 = -2.

The determinant of H is calculated as (3*4) - (1*2) = 12 - 2 = 10.

The determinant of H is (3*4) - (2*1) = 12 - 2 = 10.

Determinant of H = (3*5) - (2*1) = 15 - 2 = 13.