### Video Transcript

Given that ๐ฆ is equal to six
๐ฅ to the power of five plus three ๐ฅ squared minus seven ๐ฅ plus six, determine
the second derivative of ๐ฆ with respect to ๐ฅ.

Here weโve been given a
function in ๐ฅ. And weโre being asked to find d
two ๐ฆ by d๐ฅ squared. To do this, weโll differentiate
once to find d๐ฆ by d๐ฅ then differentiate that once again to find the second
derivative. We recall that we can
differentiate a function of the form ๐๐ฅ to the power of ๐ with respect to ๐ฅ
for some constant rational number ๐, which is not equal to zero and some
constant ๐. And we get ๐๐ times ๐ฅ to the
power of ๐ minus one. In other words, we borrow the
power of ๐ฅ and we make it the coefficient of the derivative. And then, we subtract one from
the power.

In this special case where ๐
is equal to zero, we actually have a constant. Letโs call that ๐. And the derivative of a
constant is zero. This is going to help us find
the first derivative of our function. The derivative of six ๐ฅ to the
power of five is going to be five times six ๐ฅ. And then, we subtract one from
the power. Five minus one is four. Thatโs 30๐ฅ to the fourth
power. Weโll repeat this to
differentiate three ๐ฅ squared. Itโs going to be two times
three ๐ฅ. And then, we subtract one from
the power. Two minus one is one. So, the derivative of three ๐ฅ
squared with respect to ๐ฅ is six ๐ฅ.

The derivative of negative
seven ๐ฅ is one times negative seven ๐ฅ to the power of zero. Well, thatโs just negative
seven. And, of course, six is a
constant, so the derivative of six is zero. d๐ฆ by d๐ฅ then, the first
derivative of our equation, is 30๐ฅ to the power of four plus six ๐ฅ minus
seven. Weโll differentiate each part
of this expression once more to find the second derivative.

Weโll do it piece-by-piece. The derivative of 30๐ฅ to the
power of four is four times 30๐ฅ to the power of three. The derivative of six ๐ฅ is
six. And the derivative of negative
seven is zero. Simplifying fully, and we see
that the second derivative of ๐ฆ with respect to ๐ฅ is 120๐ฅ cubed plus six.

In this example, we saw how we
could apply repeated differentiation to help us find the second derivative of a
polynomial function.