### Video Transcript

Given that π¦ is equal to negative
four π₯ plus seven multiplied by negative seven π₯ squared minus four, determine d
two π¦ by dπ₯ squared.

The question tells us that π¦ is
equal to the product of two polynomials, and it wants us to determine d two π¦ by
dπ₯ squared. And we know this means the second
derivative of π¦ with respect to π₯. Weβre going to need to
differentiate this expression twice. Since weβre given π¦ as the product
of two functions, we might be tempted to use the product rule, and this would
work. However, since our factors are just
polynomials with two terms each, itβs actually simpler in this case to just multiply
out our factors.

Since weβll just get a polynomial
with four terms, weβll multiply these together by using the FOIL method. Letβs start by multiplying the
first two terms. We get negative four times negative
seven π₯ squared is equal to 28π₯ cubed. Next, we need to multiply our outer
terms. This gives us negative four π₯
times negative four, which is equal to 16π₯. Next, we want to multiply our inner
two terms. This gives us seven multiplied by
negative seven π₯ squared, which is equal to negative 49π₯ squared. Finally, we want to multiply the
last two terms in our factors. We get seven times negative four,
which is equal to negative 28.

So, we found the following
expression for π¦. Itβs a polynomial with four
terms. Letβs switch our middle two terms
around so weβre writing it in decreasing exponents of π₯. This gives us π¦ is equal to 28π₯
cubed minus 49π₯ squared plus 16π₯ minus 28. Remember, the question wants us to
find the second derivative of π¦ with respect to π₯. Weβll start by finding the first
derivative of π¦ with respect to π₯. Thatβs dπ¦ by dπ₯, which is equal
to the derivative of 28π₯ cubed minus 49π₯ squared plus 16π₯ minus 28 with respect
to π₯.

But this is just the derivative of
a polynomial. We can do this term by term by
using the power rule for differentiation, which tells us for constants π and π,
the derivative of ππ₯ to the πth power with respect to π₯ is equal to π times π
times π₯ to the power of π minus one. We multiply by the exponent of π₯
and reduce this exponent by one. Using this, weβll differentiate our
first term. We get three times 28 times π₯ to
the power of three minus one. This simplifies to give us 84π₯
squared. We can also use this to
differentiate our second term. We get negative 49 times two, which
is negative 98. And then, we have π₯ to the power
of two minus one, which is π₯.

We could also differentiate our
last two terms by using the power rule for differentiation. However, itβs simpler to notice
that these two terms make a linear function. So, their slope will be the
coefficient of π₯, which is 16. So, we found an expression for dπ¦
by dπ₯. We can use this to find an
expression for our second derivative of π¦ with respect to π₯. Our second derivative of π¦ with
respect to π₯ would just be the derivative of dπ¦ by dπ₯ with respect to π₯.

So, to find d two π¦ by dπ₯
squared, weβre going to differentiate dπ¦ by dπ₯ with respect to π₯. Thatβs the derivative of 84π₯
squared minus 98π₯ plus 16 with respect to π₯. And again, this is the derivative
of a polynomial. So, we can do this by using the
power rule for differentiation. Our first term will be two times 84
times π₯ to the first power, which is 168π₯. And our second term will just be
the coefficient of π₯, which is 98.

Therefore, weβve shown if π¦ is
equal to negative four π₯ plus seven times negative seven π₯ squared minus four,
then d two π¦ by dπ₯ squared is equal to 168π₯ minus 98.