Final answer:
The derivative of the function k(x) = -4(10x^2 + 7) - 6 is found by applying the power rule of differentiation and the fact that the derivative of a constant is zero. The resulting derivative is k'(x) = -80x.
Step-by-step explanation:
To find the derivative of the function k(x) = -4(10x^2 + 7) - 6, we use the basic rules of differentiation. Firstly, recall that the derivative of a constant is zero and the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
We apply the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1). Therefore, the derivative of -4(10x^2) is -4 times the derivative of 10x^2, which is 20x based on the power rule. Hence, we get -80x.
Combining everything, we differentiate -4(10x^2 + 7) term by term and subtract the derivative of the constant 6 to get:
k'(x) = -80x - 0 - 0 = -80x.
So, the derivative of the function k(x) is -80x.