# Video: Differentiating Polynomials Using the Chain Rule

Determine the derivative of 𝑦 = (−2𝑥² − 3𝑥 + 4)⁵⁵.

01:20

### Video Transcript

Determine the derivative of 𝑦 equals negative two 𝑥 squared minus three 𝑥 plus four to the power of 55.

Now this is where we really see the importance of the chain rule. When we have an exponent as high as 55, we certainly don’t want to attempt to distribute all the parentheses. Instead, we’re going to use the chain rule extension of the power rule, which tells us that the derivative of 𝑓 of 𝑥 to the 𝑛 is 𝑓 prime of 𝑥 multiplied by 𝑛 multiplied by 𝑓 of 𝑥 to the 𝑛 minus one.

So, 𝑓 of 𝑥 will be that function inside the parentheses, negative two 𝑥 squared minus three 𝑥 plus four. We can apply the power rule to differentiate 𝑓 of 𝑥, giving negative four 𝑥 minus three. Now we can work out d𝑦 by d𝑥. It’s equal to 𝑓 prime of 𝑥, that’s negative four 𝑥 minus three, multiplied by 𝑛, that’s 55, multiplied by 𝑓 of 𝑥 to the power of 𝑛 minus one, that’s negative two 𝑥 squared minus three 𝑥 plus four to the power of 54.

There’s no need to expand the parentheses. So, we’ve found that d𝑦 by d𝑥 is equal to 55 multiplied by negative four 𝑥 minus three multiplied by negative two 𝑥 squared minus three 𝑥 plus four to the power of 54. And we’ve done this by applying the chain rule extension to the power rule.