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 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 dimension of electric charge is [M^1 L^2 T^-3 I^1].

Frequency is defined as the number of cycles per unit time, thus its dimension is [M^0L^0T^-1].

The gravitational constant G has dimensions of [M^-1L^3T^-2] as it relates mass, distance, and time in the law of gravitation.

The dimensional formula for acceleration is [M^0 L^1 T^-2], as it is defined as the change in velocity per unit time.

The dimensional formula for electric charge is [M^1 L^2 T^-3 I^-1], derived from the definition of current (I = Q/t).

Energy has the dimensional formula [M^1 L^2 T^-2], as it is measured in Joules (1 J = 1 kg·m²/s²).

The dimensional formula for frequency is [M^0 L^0 T^-1], as it is defined as the number of cycles per unit time.

Pressure is defined as force per unit area. The dimensional formula for force is M¹L¹T⁻², and for area is L², thus pressure = M¹L¹T⁻² / L² = M¹L⁻¹T⁻².

Velocity is defined as displacement per unit time, which gives the dimensional formula of [MLT⁻¹].

Work has the dimensional formula [M^1 L^2 T^-2].

The electric field due to a negative charge points towards the charge.