Video Transcript
If 𝑦 is equal to the fifth root of
five 𝑥 plus two to the seventh power, find d𝑦 by d𝑥.
The question wants us to find d𝑦
by d𝑥. That’s the first derivative of 𝑦
with respect to 𝑥. And we can see that 𝑦 is equal to
the fifth root of the seventh power of a linear function. That’s the composition of three
functions. So we could do this by using the
chain rule twice. And this would work. However, there’s a simpler method
by using our laws of exponents. We’ll use the following two
versions of our laws of exponents.
First, we know the fifth root of 𝑎
is just equal to 𝑎 to the power of one-fifth. Next, we know that 𝑎 to the power
of 𝑏 all raised to the power of 𝑐 is just equal to 𝑎 to the power of 𝑏 times
𝑐. Using these, we can show that 𝑦 is
equal to five 𝑥 plus two raised to the power of seven over five. But now, we can see that 𝑦 is the
composition of only two functions. It’s a linear function raised to
the power of seven over five. So we can differentiate this by
using the chain rule. We’ll set our inner function five
𝑥 plus two to be equal to 𝑢. This gives us that 𝑦 is equal to
𝑢 to the power of seven over five.
We now recall the following version
of the chain rule. If we have 𝑦 is a function of 𝑢
and 𝑢 is a function of 𝑥, then we can find the derivative of 𝑦 with respect to 𝑥
by first finding the derivative of 𝑦 with respect to 𝑢 and multiplying this by the
derivative of 𝑢 with respect to 𝑥. And this is exactly what we
have. We have that 𝑦 is a function of
𝑢. It’s 𝑢 to the power of seven over
five. And we have that 𝑢 is a function
of 𝑥. It’s five 𝑥 plus two. So we need to find d𝑦 by d𝑢 and
d𝑢 by d𝑥.
Let’s start by finding d𝑦 by
d𝑢. That’s the derivative of 𝑢 to the
power of seven over five with respect to 𝑢. And we can do this by using the
power rule for differentiation. We multiply by the exponent of 𝑢,
that’s seven over five, and reduce this exponent by one. This gives us seven-fifths times 𝑢
to the power of seven-fifths minus one. And seven-fifths minus one is equal
to two-fifths. So we found an expression for d𝑦
by d𝑢. Let’s now find an expression for
d𝑢 by d𝑥. We have that d𝑢 by d𝑥 is the
first derivative of 𝑢 with respect to 𝑥.
And remember, 𝑢 is equal to five
𝑥 plus two. So we want to differentiate five 𝑥
plus two with respect to 𝑥. And we can evaluate this by using
the power rule for differentiation. Or we can notice that this is a
linear function. So its slope is equal to the
coefficient of 𝑥, which is five. So we found expressions for d𝑦 by
d𝑢 and d𝑢 by d𝑥. We can now use the chain rule. The chain rule tells us d𝑦 by d𝑥
is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. And we found that d𝑦 by d𝑢 is
equal to seven-fifths 𝑢 to the power of two over five. And d𝑢 by d𝑥 is equal to
five. And we can simplify this by
canceling the shared factor of five in the numerator and the denominator, giving us
seven times 𝑢 to the power of two over five.
But remember, we’re finding an
expression for the derivative of 𝑦 with respect to 𝑥. So we want our answer to be in
terms of 𝑥. So we’ll use our substitution 𝑢 is
equal to five 𝑥 plus two. So by using the substitution 𝑢 is
equal to five 𝑥 plus two, we’ve shown that d𝑦 by d𝑥 is equal to seven times five
𝑥 plus two to the power of two-fifths. And we could leave our answer like
this. However, remember, our original
expression for 𝑦 contained the fifth root of a linear function. So we can use both of our laws of
exponents to write our answer in this form. Doing this, we get d𝑦 by d𝑥 is
equal to seven times the fifth root of five 𝑥 plus two squared.
Therefore, by using the chain rule,
we’ve shown if 𝑦 is equal to the fifth root of five 𝑥 plus two to the seventh
power, then d𝑦 by d𝑥 is equal to seven times the fifth root of five 𝑥 plus two
squared.
