Can you help me find the derivative of this function, I keep messing up of the steps :(

f(x)= (x-2)(x^2-3x-1)

(x-2)(2x-3) +(x^2-3x-1)*1

Of course! I can help you with that. To find the derivative of the function f(x) = (x-2)(x^2-3x-1), we can use the product rule. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:

(d/dx)[u(x)*v(x)] = u(x)*v'(x) + u'(x)*v(x)

Let's begin by finding the derivative of each term separately:

The first term, u(x) = (x-2), has a derivative of u'(x) = 1.

The second term, v(x) = (x^2-3x-1), requires the application of the power rule and constant rule:

- The derivative of x^2 is 2x.
- The derivative of -3x is -3.
- The derivative of -1 is 0, since it's a constant.

Now, let's find v'(x) = 2x - 3 + 0 = 2x - 3.

Finally, we can apply the product rule using the derivatives we found:

f'(x) = u(x)*v'(x) + u'(x)*v(x)
= (x-2)(2x - 3) + 1(x^2-3x-1)
= 2x^2 - 3x - 4x + 6 + x^2 - 3x - 1
= 3x^2 - 10x + 5

Therefore, the derivative of the function f(x) = (x-2)(x^2-3x-1) is f'(x) = 3x^2 - 10x + 5.

Yes, I'd be happy to help you find the derivative of the function f(x) = (x-2)(x^2-3x-1). The derivative is a mathematical operation that gives us the rate of change of a function at any given point. In calculus, we use various rules and techniques to find derivatives.

To find the derivative of the given function, f(x) = (x-2)(x^2-3x-1), we can use the product rule, which states that the derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function.

Let's break it down step-by-step:

1. Expand the function: f(x) = (x-2)(x^2-3x-1)
Simplify: f(x) = x^3 - 4x^2 + 7x - 2x^2 + 6x + 2

2. Combine like terms: f(x) = x^3 - 6x^2 + 13x + 2

3. Apply the product rule:
Let's call the first function u(x) = (x - 2) and the second function v(x) = (x^2 - 3x - 1).
Then, the derivative of u(x) is du(x)/dx = 1, and the derivative of v(x) is dv(x)/dx = 2x - 3.

4. Apply the product rule: f'(x) = u(x) * dv(x)/dx + v(x) * du(x)/dx
f'(x) = (x - 2)(2x - 3) + (x^2 - 3x - 1)(1)

5. Simplify and collect like terms: f'(x) = 2x^2 - 3x - 4x + 6 + x^2 - 3x - 1
f'(x) = 3x^2 - 10x + 5

So, the derivative of f(x) = (x-2)(x^2-3x-1) is f'(x) = 3x^2 - 10x + 5.

Remember to double-check your work and practice similar problems to reinforce your understanding of the derivative rules and techniques.