### Video Transcript

Given that π¦ is equal to six
π₯ to the power of five plus three π₯ squared minus seven π₯ plus six, determine
the second derivative of π¦ with respect to π₯.

Here weβve been given a
function in π₯. And weβre being asked to find d
two π¦ by dπ₯ squared. To do this, weβll differentiate
once to find dπ¦ by dπ₯ then differentiate that once again to find the second
derivative. We recall that we can
differentiate a function of the form ππ₯ to the power of π with respect to π₯
for some constant rational number π, which is not equal to zero and some
constant π. And we get ππ times π₯ to the
power of π minus one. In other words, we borrow the
power of π₯ and we make it the coefficient of the derivative. And then, we subtract one from
the power.

In this special case where π
is equal to zero, we actually have a constant. Letβs call that π. 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 π₯ to the
power of five is going to be five times six π₯. And then, we subtract one from
the power. Five minus one is four. Thatβs 30π₯ to the fourth
power. Weβll repeat this to
differentiate three π₯ squared. Itβs going to be two times
three π₯. And then, we subtract one from
the power. Two minus one is one. So, the derivative of three π₯
squared with respect to π₯ is six π₯.

The derivative of negative
seven π₯ is one times negative seven π₯ to the power of zero. Well, thatβs just negative
seven. And, of course, six is a
constant, so the derivative of six is zero. dπ¦ by dπ₯ then, the first
derivative of our equation, is 30π₯ to the power of four plus six π₯ minus
seven. Weβll differentiate each part
of this expression once more to find the second derivative.

Weβll do it piece-by-piece. The derivative of 30π₯ to the
power of four is four times 30π₯ to the power of three. The derivative of six π₯ is
six. And the derivative of negative
seven is zero. Simplifying fully, and we see
that the second derivative of π¦ with respect to π₯ is 120π₯ 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.