Use the product rule to determine the derivative of $f(x)=(x^2-3)(5x^3-7)(-10x^4+8)$.

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Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

Use the product rule to determine the derivative of $f(x)=(x^2-3)(5x^3-7)(-10x^4+8)$.

Answer: 

$f'(x)=-450x^8+1050x^6+420x^5+200x^4-840x^3-360x^2-112x$

SAMPLE SOLUTION

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Solution: 

We are explicitly asked to use the Product Rule. But there are three functions multiplied together and not two like we normally see for the product rule. There are a couple of ways to do this and they lead to the same answer. We can define <Sign in to see all the formulas> and <Sign in to see all the formulas> and apply the product rule to $h(x)$ and $g(x)$. Or we could define <Sign in to see all the formulas> and <Sign in to see all the formulas> and apply the product rule. Of course when we need to find $h'(x)$ or $g'(x)$ we will use the product rule again there. We will do the first method where <Sign in to see all the formulas> and <Sign in to see all the formulas>.

Before we get started let's note that we are dealing with polynomials and we know what the derivative of a polynomial is. Therefore,

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So here are the steps:

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