Video Transcript
Determine the value of the
second derivative of the function 𝑦 equals 12𝑥 minus eight over 𝑥 at one,
four.
We have an equation for 𝑦 in
terms of 𝑥. And we’re being asked to find
the value of the second derivative at the point with Cartesian coordinates one,
four. We’ll begin then by simply
finding an expression for the second derivative. To do this, we’ll differentiate
our function once to find d𝑦 by d𝑥 then differentiate with respect to 𝑥 once
more. It might help, before we do, to
write 𝑦 as 12𝑥 minus eight 𝑥 to the negative one. Then, we differentiate as
normal.
The derivative of 12𝑥 with
respect to 𝑥 is simply 12. And the derivative of negative
eight 𝑥 to the power of negative one is negative negative one times eight 𝑥 to
the power of negative two. And be really careful. A common mistake here is to
spot the negative one and think that when we subtract one, we get to zero. When we simplify this, we see
that the first derivative is 12 plus eight 𝑥 to the negative two. Let’s repeat this process to
find the second derivative.
The derivative of 12 is
zero. And then, when we differentiate
eight 𝑥 to the negative two with respect to 𝑥, we get negative two times eight
𝑥 to the negative three. That’s negative 16𝑥 to the
negative three. And, of course, we can change
this back to negative 16 over 𝑥 cubed if we like.
We need to determine the value
of the second derivative at one, four. This is a Cartesian
coordinate. It has an 𝑥-value of one and a
𝑦-value of four. So, we’re going to substitute
𝑥 is equal to one into our equation for the second derivative. This gives us negative 16 over
one cubed, which is negative 16.
