Question

Find the derivative using Power Rule and Product Ruley=(x^4+3)(-4x^5+5x^4+5)

Answer 1

To determine the derivative using Power Rule and Product Rule

$y=(x^4+3)(-4x^5+5x^4+5)$

Applying the product rule of differentiation:

Let f rep x^4+3

Let g rep -4x^5+5x^4+5

$\begin{gathered} (f.g)^1=f^1.g+g^1.f \\ f=x^4+3,\: g=-4x^5+5x^4+5 \end{gathered}$

$\begin{gathered} \frac{d}{dx}(\mleft(x^4+3\mright)\mleft(-4x^5+5x^4+5\mright)= \\ \frac{d}{dx}\mleft(x^4+3\mright)\mleft(-4x^5+5x^4+5\mright)+\frac{d}{dx}(-4x^5+5x^4+5)(x^4+3) \end{gathered}$$\begin{gathered} \frac{d}{dx}(x^4+3)=4x^3 \\ \frac{d}{dx}(-4x^5+5x^4+5)=-20x^4+20x^3 \\ 4x^3\mleft(-4x^5+5x^4+5\mright)+\mleft(-20x^4+20x^3\mright)\mleft(x^4+3\mright) \end{gathered}$

Simplifying the expression

$\begin{gathered} 4x^3(-4x^5+5x^4+5)+(-20x^4+20x^3)(x^4+3) \\ -16x^8+20x^7+20x^3-20x^8-60x^4+20x^7+60x^3 \\ collect\text{ like terms} \\ -16x^8-20x^8+20x^7+20x^7-60x^4+20x^3+60x^3 \\ -36x^8+40x^7-60x^4+80x^3 \end{gathered}$

Hence the final answer is - 36x^8 + 40x^7 - 60x^4 + 80x^3