Video Transcript
Given that the function 𝑓 of 𝑥 is
equal to 𝑎𝑥 cubed plus eight 𝑥 squared and 𝑓 triple prime of 𝑥 is equal to
nine, find the value of 𝑎.
We’re given a function 𝑓 of
𝑥. And we’re told that the third
derivative of 𝑓 of 𝑥 with respect to 𝑥 is equal to nine. We need to use this information to
find the value of the constant 𝑎. First, since our value of 𝑎 is a
constant, we can see that 𝑓 of 𝑥 is a cubic polynomial. And we know how to find the third
derivative of a cubic polynomial. We need to differentiate 𝑓 of 𝑥
three times with respect to 𝑥. And we can do this by using the
power rule for differentiation. So the first thing we’re going to
need to do is differentiate this with respect to 𝑥 to find an expression for 𝑓
prime of 𝑥.
To do this, we need to recall the
power rule for differentiation. For any real constants 𝑏 and 𝑛,
the derivative of 𝑏𝑥 to the 𝑛th power with respect to 𝑥 is equal to 𝑏 times 𝑛
times 𝑥 to the power of 𝑛 minus one. We multiply by our exponent of 𝑥
and then reduce this exponent by one. We can apply this term by term to
differentiate 𝑓 of 𝑥. In our first term, the exponent 𝑛
is equal to three. So we need to multiply by three and
reduce this exponent by one. This gives us 𝑎 times three 𝑥
squared. In our second term, our exponent 𝑛
is equal to two. So we need to multiply it by two
and then reduce this exponent by one, giving us eight times two 𝑥 to the first
power.
And of course we can simplify
this. 𝑎 times three 𝑥 squared is equal
to three 𝑎𝑥 squared, and eight times two 𝑥 to the first power is equal to
16𝑥. However, the information we’re
given is about the third derivative of 𝑓 of 𝑥. So we’re going to need to
differentiate this two more times. We’ll do this once again by using
the power rule for differentiation. The exponent of 𝑥 in our first
term is two. So we need to multiply by two and
reduce this exponent by one. This gives us three 𝑎 times two 𝑥
to the first power. In our second term, we can write 𝑥
as 𝑥 to the first power. Then we can differentiate this by
using the power rule for differentiation. Our exponent 𝑥 is one, so we get
16 times one 𝑥 to the zeroth power.
Then we can simplify this. We know 𝑥 to the first power is
equal to 𝑥, and 𝑥 to the zeroth power is equal to one. So this simplifies to give us six
𝑎𝑥 plus 16. We’re now ready to find an
expression for 𝑓 triple prime of 𝑥. We just need to differentiate our
expression for 𝑓 double prime of 𝑥 with respect to 𝑥. And we’ll do this term by term. Our first term can be rewritten as
six 𝑎𝑥 to the first power. We can then differentiate this by
using the power rule for differentiation. We get six 𝑎 times one 𝑥 to the
zeroth power. And there’s a few different ways of
differentiating our second term. We could use the power rule for
differentiation. However, this is a constant, so it
doesn’t vary as 𝑥 changes. So its rate of change with respect
to 𝑥 will be equal to zero.
And we can then simplify our
expression for 𝑓 triple prime of 𝑥. 𝑥 to the zeroth power is equal to
one, so we just get six 𝑎. And in the question, we’re told
that 𝑓 triple prime of 𝑥 is equal to nine. So we must have that six 𝑎 is
equal to nine. Finally, we can divide through by
six to see that 𝑎 is equal to nine over six, which is equal to three over two. Therefore, by using the power rule
for differentiation, we were able to show if 𝑓 of 𝑥 is equal to 𝑎𝑥 cubed plus
eight 𝑥 squared and 𝑓 triple prime of 𝑥 is equal to nine, then the value of the
constant 𝑎 must be three over two.
