### Video Transcript

Determine the derivative of ๐ฆ
equals negative two ๐ฅ squared minus three ๐ฅ 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 donโt want to attempt to distribute all the
parentheses. Instead, weโre going to use the
chain rule extension of the power rule, which tells us that the derivative of ๐
of ๐ฅ to the ๐ is ๐ prime of ๐ฅ multiplied by ๐ multiplied by ๐ of ๐ฅ to the
๐ minus one.

So, ๐ of ๐ฅ will be that
function inside the parentheses, negative two ๐ฅ squared minus three ๐ฅ plus
four. We can apply the power rule to
differentiate ๐ of ๐ฅ, giving negative four ๐ฅ minus three. Now we can work out d๐ฆ by
d๐ฅ. Itโs equal to ๐ prime of ๐ฅ,
thatโs negative four ๐ฅ minus three, multiplied by ๐, thatโs 55, multiplied by
๐ of ๐ฅ to the power of ๐ minus one, thatโs negative two ๐ฅ squared minus
three ๐ฅ plus four to the power of 54.

Thereโs no need to expand the
parentheses. So, weโve found that d๐ฆ by d๐ฅ
is equal to 55 multiplied by negative four ๐ฅ minus three multiplied by negative
two ๐ฅ squared minus three ๐ฅ plus four to the power of 54. And weโve done this by applying
the chain rule extension to the power rule.