To find the derivative of the function $f(x) = x^7 (8x + 4)$, we’ll use the product rule of differentiation. The product rule states that if $f(x) = g(x) \cdot h(x)$, then $f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$.

Let's define:

• $g(x) = x^7$
• $h(x) = 8x + 4$

Then, we’ll find $g'(x)$ and $h'(x)$ and apply the product rule.

Step-by-Step Solution

1. Differentiate $g(x) = x^7$: $g'(x) = 7x^6$

2. Differentiate $h(x) = 8x + 4$: $h'(x) = 8$

3. Apply the product rule: $f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$ Substitute $g(x)$, $g'(x)$, $h(x)$, and $h'(x)$: $f'(x) = (7x^6)(8x + 4) + (x^7)(8)$

4. Simplify each term:

   • For the first term: $(7x^6)(8x + 4) = 56x^7 + 28x^6$

   • For the second term: $(x^7)(8) = 8x^7$

5. Combine like terms: $f'(x) = 56x^7 + 28x^6 + 8x^7$ $= 64x^7 + 28x^6$

Final Answer

$f'(x) = 64x^7 + 28x^6$

Would you like further details, or do you have any questions?

Related Questions to Explore:

1. What if $f(x) = x^7 \cdot (8x^2 + 4)$; how would we find the derivative?
2. How does the product rule differ from the chain rule?
3. Can the solution be verified by expanding $f(x)$ first and then differentiating?
4. What is the second derivative of $f(x) = x^7 (8x + 4)$?
5. How does the power rule apply to functions like $x^7$ in differentiation?

Tip

Remember to identify and label each component of a function clearly when using the product rule to avoid confusion in complex derivatives.