Video Transcript
If 𝑥 is equal to two multiplied by
the sec of five 𝑧 and the square root of three 𝑦 is equal to the tan of five 𝑧,
find the second derivative of 𝑦 with respect to 𝑥.
Since the question asks us to find
the second derivative of 𝑦 with respect to 𝑥, it might be tempting to try and
write 𝑦 in terms of 𝑥. However, there’s a simpler method
by using the chain rule. We recall that calculating the
second derivative of 𝑦 with respect to 𝑥 is the same as differentiating d𝑦 d𝑥
with respect to 𝑥. And we recall that the chain rule
tells us that the derivative of 𝑦 with respect to 𝑥 is equal to the derivative of
𝑦 with respect to 𝑧 multiplied by the derivative of 𝑧 with respect to 𝑥. And instead of multiplying by the
derivative of 𝑧 with respect to 𝑥, since we’re given 𝑥 as a function of 𝑧, we
can do the equivalent, dividing by the derivative of 𝑥 with respect to 𝑧.
So let’s start by calculating the
derivative of 𝑥 with respect to 𝑧. That’s equal to the derivative of
two multiplied by the sec of five 𝑧 with respect to 𝑧. And we recall that the derivative
of the sec of 𝑎𝑥 with respect to 𝑥 is equal to 𝑎 multiplied by the sec of 𝑎𝑥
multiplied by the tan of 𝑎𝑥. Applying this gives us that the
derivative of 𝑥 with respect to 𝑧 is equal to two multiplied by five, which we can
simplify to 10, multiplied by the sec of five 𝑧 multiplied by the tan of five
𝑧.
Calculating the derivative of 𝑦
with respect to 𝑧 is more difficult since we’re not given 𝑦 directly as a function
of 𝑧. The simplest way to do this is to
rearrange our equation to make 𝑦 the subject. We’ll do this by squaring both
sides of the equation. Evaluating this gives us that three
𝑦 is equal to the tan squared of five 𝑧.
Next, we differentiate both sides
of our equation with respect to 𝑧. Differentiating three 𝑦 with
respect to 𝑧 gives us three multiplied by the derivative of 𝑦 with respect to
𝑧. We can calculate the derivative of
the tan of five 𝑧 all squared with respect to 𝑧. By recalling that, for a constant
𝑎, the derivative with respect to 𝑥 of the tan of 𝑎 multiplied by 𝑥 is equal to
𝑎 multiplied by the sec of 𝑎𝑥 squared.
And by using the chain rule, we
have that the derivative of the function 𝑓 of 𝑥 all squared is equal to two
multiplied by 𝑓 prime of 𝑥 multiplied by 𝑓 of 𝑥. Using this gives us that the
derivative of the tan of five 𝑧 all squared with respect to 𝑧 is equal to two
multiplied by our function 𝑓 prime of 𝑧. Which we’ve shown to be equal to
five multiplied by the sec of five 𝑧 all squared. And then we multiply this by our
function 𝑓 of 𝑧, which is the tan of five 𝑧.
We can simplify this by multiplying
the two and the five to get 10 and then dividing both sides of our equation by
three. This gives us that the derivative
of 𝑦 with respect to 𝑧 is equal to 10 over three multiplied by the sec of five 𝑧
squared multiplied by the tan of five 𝑧.
We’re now ready to apply our chain
rule that the derivative of 𝑦 with respect to 𝑥 is equal to the derivative of 𝑦
with respect to 𝑧 divided by the derivative of 𝑥 with respect to 𝑧. We have a numerator of d𝑦 d𝑧,
which is equal to ten-thirds multiplied by the sec of five 𝑧 all squared multiplied
by the tan of five 𝑧. And a denominator of d𝑥 by d𝑧,
which is equal to 10 multiplied by the sec of five 𝑧 multiplied by the tan of five
𝑧.
We can then start canceling the
shared factors. We can cancel the shared factor of
the tan of five 𝑧 in the numerator and the denominator. We can cancel one shared factor of
the sec of five 𝑧 in the numerator and the denominator. And we can cancel the shared factor
of 10 in the numerator and the denominator.
Therefore, we’ve shown that the
derivative of 𝑦 with respect to 𝑥 is equal to the sec of five 𝑧 all divided by
three. So we’ve shown that our derivative
function d𝑦 d𝑥 is a function of 𝑧. Let’s call this 𝑓 of 𝑧. And the question wants us to
calculate the second derivative of 𝑦 with respect to 𝑥. And that’s the same as
differentiating our function 𝑓 of 𝑧 with respect to 𝑥.
And we’ve already seen how to do
this. We can use the chain rule. This gives us the derivative of 𝑓
with respect to 𝑥 is equal to the derivative of 𝑓 with respect to 𝑧 multiplied by
the derivative of 𝑧 with respect to 𝑥. And just as we did before, instead
of multiplying by the derivative of 𝑧 with respect to 𝑥, since we’re given 𝑥 as a
function of 𝑧, we can instead divide by the derivative of 𝑥 with respect to
𝑧.
So what we have is the second
derivative of 𝑦 with respect to 𝑥 is equal to the derivative of d𝑦 d𝑥 with
respect to 𝑥. And we’ve called d𝑦 d𝑥 𝑓 of
𝑧. And then using the chain rule,
we’ve shown that this is equal to d𝑓 d𝑧 divided by d𝑥 d𝑧. Substituting in our expression for
𝑓, we get the derivative of 𝑓 with respect to 𝑧 is equal to the derivative of the
sec of five 𝑧 all divided by three with respect to 𝑧.
Then we can substitute in our
expression for d𝑥 by d𝑧. This is 10 sec of five 𝑧
multiplied by the tan of five 𝑧. We can differentiate the sec of
five 𝑧 to be equal to five multiplied by the sec of five 𝑧 multiplied by the tan
of five 𝑧. And then we divide through by our
constant of three. And instead of dividing, we’re
going to multiply by the reciprocal.
Simplifying this gives us five
multiplied by the sec of five 𝑧 multiplied by the tan of five 𝑧 all divided by 30
multiplied by the sec of five 𝑧 multiplied by the tan of five 𝑧. We can then cancel the shared
factor of the tan of five 𝑧 in our numerator and our denominator and the shared
factor of the sec of five 𝑧 in the numerator and the denominator. And if we then cancel the shared
factor of five, we get one-sixth.
So we’ve shown that if 𝑥 is equal
to two multiplied by the sec of five 𝑧 and the square root of three 𝑦 is equal to
the tan of five 𝑧, then the second derivative of 𝑦 with respect to 𝑥 is equal to
one-sixth.
