Question

Find the derivative of the function.

g(x) = 6x⁻³ + 2x⁻⁵

a. g′(x) = -18x⁻³ - 10x⁻⁵
b. g′(x) = 6x⁻³ + 2x⁻⁵
c. g′(x) = 18x⁻⁴ + 10x⁻⁶

Answer 1

The derivative of the function g(x) = 6x⁻³ + 2x⁻⁵ is c.
`\(g'(x) = 18x^(-4) + 10x^(-6)\)`. Thus, the correct answer is option c.
`\(g'(x) = 18x^(-4) + 10x^(-6)\)`.

Step-by-step explanation:

To find the derivative of the given function
`\(g(x) = 6x^(-3) + 2x^(-5)\)`, we'll apply the power rule of differentiation. The power rule states that if
`\(f(x) = ax^n\)`, then
`\(f'(x) = nax^((n-1))\)`.

For the first term
`\(6x^(-3)\)`, applying the power rule gives
`\(g_1'(x) = 18x^(-4)\)`. Similarly, for the second term
`\(2x^(-5)\)`, the derivative is
`\(g_2'(x) = 10x^(-6)\)`. Adding these derivatives, we get the overall derivative of the function:

`\[ g'(x) = g_1'(x) + g_2'(x) = 18x^(-4) + 10x^(-6) \]`

Therefore, the correct derivative is
`\(g'(x) = 18x^(-4) + 10x^(-6)\)`, which corresponds to option c.

In this case, it's important to note the negative signs that arise from the power rule when differentiating terms with negative exponents. The power rule simplifies the process of finding derivatives for functions with monomials, allowing us to determine the derivative of each term separately and then sum them up. This method ensures accuracy and efficiency in finding the derivative of the entire function.