Use the product rule to determine the derivative of f(x)=(x2−3)(5x3−7)(−10x4+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)=(x2−3)(5x3−7)(−10x4+8).

Answer: 

f′(x)=−450x8+1050x6+420x5+200x4−840x3−360x2−112x

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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