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^(-1)(x) is 1/√(1-x^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 is calculated as (0*0) - (1*1) = 0 - 1 = -1.

The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).

The determinant is 0 because the rows are linearly dependent.

The determinant of the identity matrix is 1.

The determinant is calculated as (1*1) - (1*1) = 1 - 1 = 0.

The determinant is calculated as \( 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1 \).

The determinant is 0 because the rows are linearly dependent.

The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).

The determinant is calculated as \( 1*5 - 2*3 = 5 - 6 = -1 \).

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

The determinant is calculated as \( 5*8 - 6*7 = 40 - 42 = -2 \).

Calculating gives a determinant of -3.

The standard form of a parabola is (y - k)^2 = 4p(x - h). Here, p = 2, so the directrix is x = -2.

For the parabola y^2 = 4px, here 4p = 8, so p = 2. The directrix is x = -p = -2.

The distance between centers (1, 2) and (-3, -4) is calculated using the distance formula.

The distance between the foci is 6, calculated using the formula 2c where c = √(a^2 - b^2).

The distance between the foci is given by 2c, where c = √(a^2 - b^2) = √(25 - 16) = 3, so the total distance is 2c = 6.

Distance = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5.

Distance = √[(7-3)² + (1-4)²] = √[16 + 9] = √25 = 5.

Distance = √((4-1)² + (5-2)² + (6-3)²) = √(9 + 9 + 9) = 3√3.

Distance = |3(1) + 4(2) - 10| / √(3² + 4²) = |3 + 8 - 10| / 5 = 1.

Distance = |2(3) + 3(4) - 6| / √(2² + 3²) = |6 + 12 - 6| / √13 = 12/√13.

The function is undefined at x = 2, so the domain is all real numbers except 2.