Larson Calculus 5.4 #40: Derivative of y = xe^(4x) with the Product Rule
Larson Calculus 5.4 #40: Derivative of y = xe^(4x) with the Product Rule

### Video Transcript

Given that 𝑦 is equal to negative
four 𝑥 plus seven multiplied by negative seven 𝑥 squared minus four, determine d
two 𝑦 by d𝑥 squared.

The question tells us that 𝑦 is
equal to the product of two polynomials, and it wants us to determine d two 𝑦 by
d𝑥 squared. And we know this means the second
derivative of 𝑦 with respect to 𝑥. We’re going to need to
differentiate this expression twice. Since we’re given 𝑦 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, it’s actually simpler in this case to just multiply
out our factors.

Since we’ll just get a polynomial
with four terms, we’ll multiply these together by using the FOIL method. Let’s start by multiplying the
first two terms. We get negative four times negative
seven 𝑥 squared is equal to 28𝑥 cubed. Next, we need to multiply our outer
terms. This gives us negative four 𝑥
times negative four, which is equal to 16𝑥. Next, we want to multiply our inner
two terms. This gives us seven multiplied by
negative seven 𝑥 squared, which is equal to negative 49𝑥 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 𝑦. It’s a polynomial with four
terms. Let’s switch our middle two terms
around so we’re writing it in decreasing exponents of 𝑥. This gives us 𝑦 is equal to 28𝑥
cubed minus 49𝑥 squared plus 16𝑥 minus 28. Remember, the question wants us to
find the second derivative of 𝑦 with respect to 𝑥. We’ll start by finding the first
derivative of 𝑦 with respect to 𝑥. That’s d𝑦 by d𝑥, which is equal
to the derivative of 28𝑥 cubed minus 49𝑥 squared plus 16𝑥 minus 28 with respect
to 𝑥.

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 𝑎 and 𝑛,
the derivative of 𝑎𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑛 times 𝑎
times 𝑥 to the power of 𝑛 minus one. We multiply by the exponent of 𝑥
and reduce this exponent by one. Using this, we’ll differentiate our
first term. We get three times 28 times 𝑥 to
the power of three minus one. This simplifies to give us 84𝑥
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 𝑥 to the power
of two minus one, which is 𝑥.

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

So, to find d two 𝑦 by d𝑥
squared, we’re going to differentiate d𝑦 by d𝑥 with respect to 𝑥. That’s the derivative of 84𝑥
squared minus 98𝑥 plus 16 with respect to 𝑥. 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 𝑥 to the first power, which is 168𝑥. And our second term will just be
the coefficient of 𝑥, which is 98.

Therefore, we’ve shown if 𝑦 is
equal to negative four 𝑥 plus seven times negative seven 𝑥 squared minus four,
then d two 𝑦 by d𝑥 squared is equal to 168𝑥 minus 98.

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