Question Statement

Find the derivative of $y = (x^{2} - 7) (x^{2} + 4 x + 2)$ with respect to $x$ .

Background and Explanation

This problem requires the product rule for differentiation, which is essential when finding the derivative of two functions multiplied together. You should be comfortable with the power rule $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ and recognizing when to apply the product rule rather than expanding first.

Solution

We are given the function: $y = (x^{2} - 7) (x^{2} + 4 x + 2)$

Since this is a product of two polynomial functions, we apply the product rule. Let us define:

$u = x^{2} - 7$ (the first factor)
$v = x^{2} + 4 x + 2$ (the second factor)

The product rule states: $\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$

Step 1: Differentiate each factor separately using the power rule. $\frac{d u}{d x} = \frac{d}{d x} (x^{2} - 7) = 2 x - 0 = 2 x$

$\frac{d v}{d x} = \frac{d}{d x} (x^{2} + 4 x + 2) = 2 x + 4 + 0 = 2 x + 4$

Step 2: Substitute into the product rule formula. $\frac{d y}{d x} = (x^{2} - 7) (2 x + 4) + (x^{2} + 4 x + 2) (2 x)$

Or equivalently (swapping the order as in the original): $\frac{d y}{d x} = (x^{2} + 4 x + 2) (2 x) + (x^{2} - 7) (2 x + 4)$

Step 3: Expand each term. First term: $(x^{2} + 4 x + 2) (2 x) = 2 x^{3} + 8 x^{2} + 4 x$

Second term: $(x^{2} - 7) (2 x + 4)$ $= x^{2} (2 x + 4) - 7 (2 x + 4)$ $= 2 x^{3} + 4 x^{2} - 14 x - 28$

Step 4: Combine all terms. $\frac{d y}{d x} = (2 x^{3} + 8 x^{2} + 4 x) + (2 x^{3} + 4 x^{2} - 14 x - 28)$

Step 5: Group like terms.

$x^{3}$ terms: $2 x^{3} + 2 x^{3} = 4 x^{3}$
$x^{2}$ terms: $8 x^{2} + 4 x^{2} = 12 x^{2}$
$x$ terms: $4 x - 14 x = - 10 x$
Constant: $- 28$

Therefore: $\frac{d y}{d x} = 4 x^{3} + 12 x^{2} - 10 x - 28$

Key Formulas or Methods Used

Product Rule: $\frac{d}{d x} [u \cdot v] = u \frac{d v}{d x} + v \frac{d u}{d x}$ (or $\frac{d}{d x} [uv] = u^{'} v + u v^{'}$ )
Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
Constant Rule: $\frac{d}{d x} (c) = 0$ where $c$ is a constant
Sum/Difference Rule: $\frac{d}{d x} [f (x) \pm g (x)] = f^{'} (x) \pm g^{'} (x)$

Summary of Steps

Identify the two functions being multiplied: $u = x^{2} - 7$ and $v = x^{2} + 4 x + 2$
Differentiate each function separately: $u^{'} = 2 x$ and $v^{'} = 2 x + 4$
Apply the product rule formula: $\frac{d y}{d x} = u v^{'} + v u^{'}$
Substitute the expressions: $(x^{2} - 7) (2 x + 4) + (x^{2} + 4 x + 2) (2 x)$
Expand both products using distribution (FOIL/distributive property)
Combine like terms to obtain the final simplified derivative: $4 x^{3} + 12 x^{2} - 10 x - 28$

Solution

We are given the function:

y = (x^{2} - 7) (x^{2} + 4 x + 2)

Since this is a product of two polynomial functions, we apply the product rule. Let us define:

u = x^{2} - 7

(the first factor)

v = x^{2} + 4 x + 2

(the second factor)

The product rule states:

\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

Step 1: Differentiate each factor separately using the power rule.

\frac{d u}{d x} = \frac{d}{d x} (x^{2} - 7) = 2 x - 0 = 2 x

\frac{d v}{d x} = \frac{d}{d x} (x^{2} + 4 x + 2) = 2 x + 4 + 0 = 2 x + 4

Step 2: Substitute into the product rule formula.

\frac{d y}{d x} = (x^{2} - 7) (2 x + 4) + (x^{2} + 4 x + 2) (2 x)

Or equivalently (swapping the order as in the original):

\frac{d y}{d x} = (x^{2} + 4 x + 2) (2 x) + (x^{2} - 7) (2 x + 4)

Step 3: Expand each term. First term:

(x^{2} + 4 x + 2) (2 x) = 2 x^{3} + 8 x^{2} + 4 x

Second term:

(x^{2} - 7) (2 x + 4)

= x^{2} (2 x + 4) - 7 (2 x + 4)

= 2 x^{3} + 4 x^{2} - 14 x - 28

Step 4: Combine all terms.

\frac{d y}{d x} = (2 x^{3} + 8 x^{2} + 4 x) + (2 x^{3} + 4 x^{2} - 14 x - 28)

Step 5: Group like terms.

x^{3}

terms:

2 x^{3} + 2 x^{3} = 4 x^{3}

x^{2}

terms:

8 x^{2} + 4 x^{2} = 12 x^{2}

x

terms:

4 x - 14 x = - 10 x

Constant:

- 28

Therefore:

\frac{d y}{d x} = 4 x^{3} + 12 x^{2} - 10 x - 28

Summary of Steps

Identify the two functions being multiplied:

u = x^{2} - 7

and

v = x^{2} + 4 x + 2

Differentiate each function separately:

u^{'} = 2 x

and

v^{'} = 2 x + 4

Apply the product rule formula:

\frac{d y}{d x} = u v^{'} + v u^{'}

Substitute the expressions:

(x^{2} - 7) (2 x + 4) + (x^{2} + 4 x + 2) (2 x)

Expand both products using distribution (FOIL/distributive property)

Combine like terms to obtain the final simplified derivative:

4 x^{3} + 12 x^{2} - 10 x - 28

2.5 Q-2

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

2.5 Q-2

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps