Question

Given the function:

f(x)=x9−x
To find the derivative f′(x), we will use the product rule. The product rule states that if you have a function u(x)=x and a function v(x)=9−x, then the derivative of their product is:
f′(x)=u(x)⋅v′(x)+v(x)⋅u′(x)

Answer 1

To find the derivative of the function f(x) = x^9 - x using the product rule, we can use the formula f'(x) = u(x) * v'(x) + v(x) * u'(x), where u(x) = x and v(x) = 9 - x. The derivative of u(x) with respect to x is 1, and the derivative of v(x) with respect to x is -1. Therefore, the derivative of f(x) is f'(x) = -x + 9.

Step-by-step explanation:

To find the derivative of the function f(x) = x^9 - x using the product rule, we first need to find the derivatives of the functions u(x) = x and v(x) = 9 - x. The derivative of u(x) with respect to x, denoted as u'(x), is 1 since the derivative of x is 1. The derivative of v(x) with respect to x, denoted as v'(x), is -1 since the derivative of -x is -1.

Using the product rule, f'(x) = u(x) * v'(x) + v(x) * u'(x). Substituting the values, f'(x) = x * (-1) + (9 - x) * 1. Simplifying, we get f'(x) = -x + 9.