Consider the following function.

*h*(*x*) = *x*(7 + 2*x*) 

(a)

Calculate the derivative of the given function without using either the product or quotient rule.

*h'*(*x*) =  
 
(b)

Use the product or quotient rule to find the derivative. Check that you obtain the same answer.

*h'*(*x*) =  
 

We are given the function \( h(x) = x(7 + 2x) \), and we need to find its derivative in two different ways.

### (a) Calculating the derivative without using the product or quotient rule

First, let's expand the function:

\[
h(x) = x(7 + 2x) = 7x + 2x^2
\]

Now, differentiate term by term:

\[
\frac{d}{dx}(7x) = 7
\]
\[
\frac{d}{dx}(2x^2) = 4x
\]

Thus, the derivative of \( h(x) \) is:

\[
h'(x) = 7 + 4x
\]

### (b) Using the product rule to find the derivative

We will now apply the product rule, which states that if you have two functions \( u(x) \) and \( v(x) \), then:

\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]

In this case:
- \( u(x) = x \)
- \( v(x) = 7 + 2x \)

Now, find the derivatives of \( u(x) \) and \( v(x) \):

\[
u'(x) = 1
\]
\[
v'(x) = \frac{d}{dx}(7 + 2x) = 2
\]

Using the product rule:

\[
h'(x) = u'(x)v(x) + u(x)v'(x)
\]
\[
h'(x) = 1 \cdot (7 + 2x) + x \cdot 2
\]
\[
h'(x) = (7 + 2x) + 2x
\]
\[
h'(x) = 7 + 4x
\]

### Conclusion:
Both methods give the same derivative:

\[
h'(x) = 7 + 4x
\]

Would you like further details on any part of this?

---

Here are 5 relative questions to expand on this topic:

1. How can the product rule be extended to more than two functions?
2. Can the chain rule be used in a similar function, and how does it differ from the product rule?
3. What would happen if a constant term were added to the function? How would the derivative change?
4. How would you find the second derivative of \( h(x) \)?
5. How does the quotient rule differ from the product rule in terms of application?

**Tip:** Expanding a product before differentiating can simplify the derivative calculation when possible, but the product rule is essential for more complex functions!

Consider the following function.

h(x) = x(7 + 2x)

(a)

Calculate the derivative of the given function without using either the product or quotient rule.

h'(x) =

(b)

Use the product or quotient rule to find the derivative. Check that you obtain the same answer.

h'(x) =

Learn how to find the derivative of the function h(x) = x(7 + 2x) using both expansion and the product rule. This solution covers calculus concepts like derivatives and algebraic simplification. Ideal for students in Grades 11-12.

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