Final answer:
The derivative of q(x) = (8 - 3x⁵)⁴ is -60x⁴(8 - 3x⁵)³, and the derivative of f(x) = (2x)/6 is simply 1/3.
Step-by-step explanation:
To find the derivatives of the given functions, we will apply the necessary rules of differentiation. Let's start with the function q(x) = (8 - 3x⁵)⁴. To differentiate this, we use the chain rule, which states that the derivative of a composed function is the derivative of the outside function evaluated at the inside function multiplied by the derivative of the inside function. In this case, the outside function is f(u) = u⁴ and the inside function is g(x) = 8 - 3x⁵. So, we get:
d/dx [q(x)] = 4(8 - 3x⁵)³ * (-15x⁴)
which simplifies to:
-60x⁴(8 - 3x⁵)³
Now, for the function f(x) = (2x)/6, we see that it is a simple derivative that can be found using the constant multiple rule. This rule states that the derivative of a constant times a function is the constant times the derivative of the function. Thus, we have:
d/dx [f(x)] = 2/6 * d/dx [x]
Since the derivative of x with respect to x is 1, we get:
1/3