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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

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

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2 Answers

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

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

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

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

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

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

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

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