Question

Differentiate the function r(z) = z⁻4 - z^(1/2) with respect to z.

a. r'(z) = 4z⁻5 - (1/2)z^(-1/2)
b. r'(z) = -4z⁻5 - (1/2)z^(-1/2)
c. r'(z) = -4z⁻5 - z^(-1/2)
d. r'(z) = 4z⁻5 - z^(-1/2)

Answer 1

The derivative of the function r(z) = z^-4 - z^(1/2) with respect to z is r'(z) = -4z^-5 - (1/2)z^-1/2.Option B is the correct answer.

Step-by-step explanation:

To differentiate the function r(z) = z⁻⁴ - z¹⁄₂ with respect to z, we apply the power rule of differentiation. The power rule states that if you have a function of the form f(z) = z^n, where n is any real number, then the derivative of that function with respect to z is nf(z) = nz^(n-1).

Applying the power rule, the first term of the function z⁻⁴ differentiates to -4z⁻⁵. For the second term, z¹⁄₂, its derivative is -½z^(-½) or -½z⁻½.

Combining these results, the derivative of r(z), denoted as r'(z), is r'(z) = -4z⁻⁵ - ½z⁻½.

To differentiate the function r(z) = z⁻4 - z^(1/2) with respect to z, we can use the power rule of differentiation. Here are the steps:

1. Take the derivative of the first term: z⁻4. Using the power rule, the derivative is (-4)z^(-4-1) = -4z⁻⁵.
2. Take the derivative of the second term: z^(1/2). Using the power rule, the derivative is (1/2)z^((1/2)-1) = (1/2)z^(-1/2).
3. Combine the derivatives to find the derivative of the entire function: r'(z) = -4z⁻⁵ - (1/2)z^(-1/2).