calculate the derivative of $f(x)=(x - 1)(x - 7)(x - 6)$ using the appropriate rule or combination of…

Explanation:

Step1: Expand the function

[ \begin{align*} f(x)&=(x - 1)(x - 7)(x - 6)\ &=(x^2-7x - x + 7)(x - 6)\ &=(x^2-8x + 7)(x - 6)\ &=x^3-6x^2-8x^2 + 48x+7x - 42\ &=x^3-14x^2 + 55x - 42 \end{align*} ]

Step2: Differentiate term - by - term

The derivative of (x^n) is (nx^{n - 1}). The derivative of (x^3) is (3x^{2}), the derivative of (-14x^2) is (-28x), the derivative of (55x) is (55), and the derivative of the constant (-42) is (0). So (f'(x)=3x^{2}-28x + 55).

Answer:

(3x^{2}-28x + 55)