Question

Find an example of a quadratic function that can be differentiated using the product rule and provide a step-by-step solution.

Answer 1

Let's consider a quadratic function of the form \( f(x) = x^2 \) and rewrite it in a way that allows us to apply the product rule.

We can express \( f(x) = x^2 \) as a product of two functions:

\[
f(x) = x \cdot x
\]

Now, we will differentiate \( f(x) = x \cdot x \) using the product rule. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), the derivative of their product is given by:

\[
(uv)' = u'v + uv'
\]

In our case, we have:

- \( u(x) = x \)
- \( v(x) = x \)

Now, let's find the derivatives of \( u \) and \( v \):

1. \( u' = \frac{d}{dx}(x) = 1 \)
2. \( v' = \frac{d}{dx}(x) = 1 \)

Now, applying the product rule:

\[
f'(x) = u'v + uv'
\]

Substituting in our values:

\[
f'(x) = (1)(x) + (x)(1)
\]

Now simplify:

\[
f'(x) = x + x = 2x
\]

Therefore, the derivative of the quadratic function \( f(x) = x^2 \), using the product rule, is:

\[
f'(x) = 2x
\]

### Summary:
- The quadratic function we used is \( f(x) = x \cdot x \).
- We applied the product rule to differentiate it and found that \( f'(x) = 2x \).