f(x) = (px+q) (r/x+s) differentiate with respect to x

Given the function \( f(x) = (px + q) \left( \frac{r}{x} + s \right) \), we want to differentiate it with respect to \( x \).

### Step-by-Step Differentiation:

First, apply the **product rule**, which states:
\[
\frac{d}{dx} \left[ u(x) \cdot v(x) \right] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
where \( u(x) = (px + q) \) and \( v(x) = \left( \frac{r}{x} + s \right) \).

1. Differentiate \( u(x) = px + q \) with respect to \( x \):
\[
u'(x) = p
\]

2. Differentiate \( v(x) = \frac{r}{x} + s \) with respect to \( x \):
\[
v'(x) = -\frac{r}{x^2}
\]

Now, apply the product rule:
\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
Substituting the values:
\[
f'(x) = p \cdot \left( \frac{r}{x} + s \right) + (px + q) \cdot \left( -\frac{r}{x^2} \right)
\]

### Simplifying:

1. Distribute \( p \) in the first term:
\[
p \cdot \left( \frac{r}{x} + s \right) = \frac{pr}{x} + ps
\]

2. Distribute \( (px + q) \) in the second term:
\[
(px + q) \cdot \left( -\frac{r}{x^2} \right) = -\frac{prx}{x^2} - \frac{qr}{x^2}
= -\frac{pr}{x} - \frac{qr}{x^2}
\]

Finally, combine the terms:
\[
f'(x) = \frac{pr}{x} + ps - \frac{pr}{x} - \frac{qr}{x^2}
\]

The \( \frac{pr}{x} \) terms cancel out, leaving:
\[
f'(x) = ps - \frac{qr}{x^2}
\]

Thus, the derivative of the function is:
\[
f'(x) = ps - \frac{qr}{x^2}
\]

Would you like any details on a specific step or further explanation?

Here are 5 related questions:
1. What is the product rule and how is it applied?
2. Can you differentiate functions involving natural logarithms?
3. How do you apply the quotient rule for division of two functions?
4. How do implicit derivatives work in more complex cases?
5. What are common mistakes when using the chain rule?

**Tip:** Always check if terms can cancel out after applying the product or chain rule to simplify your final expression.

Explore how to differentiate the function f(x) = (px + q)(r/x + s) step-by-step using the product rule and rational function differentiation. This solution covers key calculus concepts like the product rule and simplifying derivatives, suitable for undergraduate calculus students.

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