# Derivative: Which rule to use first?

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

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!

• Product rule then chain rule. – John Habert Feb 20 '14 at 18:21
• Product and then chain rule – Sandeep Thilakan Feb 20 '14 at 18:21
• It is impossible to apply the chain rule here. – mercio Feb 20 '14 at 18:46

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?

• I tried to but can't seem to figure it out. I got 7x^6(5+8x)^3+x^7(75x+120) which isn't right – user102817 Feb 20 '14 at 18:31
• Where do you get that 75 and 120 from? I get a 24 in front of that term. See AmWhy's answer... – imranfat Feb 20 '14 at 19:56

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:$$\frac{d}{dx}(5 + 8x)^3 = 3(5+8x)^2\cdot(8) = 24(5+8x)^2$$

• You should start to think about accepting one answer to each of your questions ;-) – Namaste Feb 25 '14 at 18:01