# How to simplify derivatives

The math problem asks to find the derivative of the function $$y=(x+1)^4(x+5)^2$$

I get to the part $$(x+1)^4 \cdot 2(x+5) + (x+5)^2 \cdot 4(x+1)^3$$

How do they arrive at the answer

$$2(x+1)^3(x+5)(3x+11) ?$$

• By taking the factor $(x+1)^3(x+5)$. – Nosrati Jan 29 '17 at 5:35
• Many texts will factor or expand. Don't worry if your answer doesn't exactly match. – Sean Roberson Jan 29 '17 at 5:54
• ^ always plug and check to see if your answer is correct – Sentient Jan 29 '17 at 6:14

## 2 Answers

\begin{align}(x+1)^4 \cdot 2(x+5) + (x+5)^2 \cdot 4(x+1)^3&=(x+1)^3(x+5)\cdot\big[(x+1)2+(x+5)4\big]\\ &=(x+1)^3(x+5)\cdot\big[2x+2+4x+20 \big]\\ &=(x+1)^3(x+5)\cdot(6x+22)\\ &=(x+1)^3(x+5)\cdot 2(3x+11)\end{align}

• @Fiona Lu There are now two answers in your question. You can either accept mine or the other by simply clicking the check mark. The upward arrow is for up vote if you wish to. – Juniven Jan 29 '17 at 7:07

$$(x+1)^4 \cdot 2(x+5) + (x+5)^2 \cdot 4(x+1)^3$$

$$(x+1)^3((x+1) \cdot 2(x+5) + (x+5)^2 \cdot 4)$$

$$(x+1)^3(x+5)( (x+1)\cdot 2 + (x+5) \cdot 4)$$

$$(x+1)^3(x+5)( 2)((x+1)+ (x+5)2)$$

$$2(x+1)^3(x+5)(x+1+2x+10)$$

$$2(x+1)^3(x+5)(3x + 11)$$

• Thanks a lot for your help! – Fiona Lu Jan 29 '17 at 6:53