Question

F(x) = (1/x +1)(x-1)
Using the product rule.

Solve step by step easily

Answer 1

To differentiate the function \( F(x) = \left( \frac{1}{x} + 1 \right)(x - 1) \) using the product rule, we will follow these steps:

### Step 1: Identify the two functions for the product rule

Let:
- \( u = \frac{1}{x} + 1 \)
- \( v = x - 1 \)

### Step 2: Differentiate \( u \) and \( v \)

#### Differentiating \( u \):
\[
u = \frac{1}{x} + 1
\]
To differentiate \( u \), we find the derivative of \( \frac{1}{x} \):
\[
\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}
\]
So,
\[
\frac{du}{dx} = -\frac{1}{x^2}
\]

#### Differentiating \( v \):
\[
v = x - 1
\]
The derivative of \( v \) is straightforward:
\[
\frac{dv}{dx} = 1
\]

### Step 3: Apply the product rule

The product rule states that:
\[
\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}
\]

Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):

\[
F'(x) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}
\]

### Step 4: Substitute the derivatives into the product rule formula

Substituting the known values:
- \( u = \frac{1}{x} + 1 \)
- \( \frac{dv}{dx} = 1 \)
- \( v = x - 1 \)
- \( \frac{du}{dx} = -\frac{1}{x^2} \)

This gives us:

\[
F'(x) = \left(\frac{1}{x} + 1\right)(1) + (x - 1)\left(-\frac{1}{x^2}\right)
\]

### Step 5: Simplifying the expression

Now simplifying the expression:
\[
F'(x) = \frac{1}{x} + 1 - \frac{x - 1}{x^2}
\]

Now let's focus on simplifying \(-\frac{x - 1}{x^2}\):
\[
-\frac{x - 1}{x^2} = -\frac{x}{x^2} + \frac{1}{x^2} = -\frac{1}{x} + \frac{1}{x^2}
\]

### Step 6: Combine all parts

Now, combining everything together:
\[
F'(x) = \frac{1}{x} + 1 - \frac{1}{x} + \frac{1}{x^2}
\]

The \(\frac{1}{x}\) cancels out:
\[
F'(x) = 1 + \frac{1}{x^2}
\]

### Final result
Thus, the derivative of \( F(x) \) is:
\[
F'(x) = 1 + \frac{1}{x^2}
\]