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.
