Derivative: Which rule to use first?

Question

$f(x)=x^7(5+8x)^3$

Would I go about finding the derivative of this problem by using the chain rule first, and then the product rule? Or would I do the opposite? Step by step instructions would be immensely helpful. Thanks!

Answer 1

You need both rules, but start of with product rule: $[x^7]'*(5+8x)^3+x^7*[(5x+8)^3]'$ In that last step you need Chain Rule which result in another factor 5. Can you work it out?

Answer 2

Use the product rule first.

$$f(x)=x^7(5+8x)^3 \implies f'(x) = 7x^6(5+8x)^3 + x^7\cdot \frac{d}{dx}\left[(5 + 8x)^3\right]$$ Then, use the chain rule to evaluate the derivative of the remaining factor above:$$\frac{d}{dx}(5 + 8x)^3 = 3(5+8x)^2\cdot(8) = 24(5+8x)^2$$