Video Transcript
Find d𝑦 by d𝑥, given that 𝑦 is
equal to the natural log of two 𝑥 plus seven cubed.
In this question, we have a
function of a function, in other words, a composite function. We can use the chain rule to find
the derivative of a composite function, though since we’re working with the natural
logarithm, it’s sensible to first consider whether there is anything we can do to
manipulate our expression before differentiating. We recall the general result for
the natural logarithm of a power. The natural logarithm of 𝑥 to the
𝑦th power is 𝑦 times the natural logarithm of 𝑥. And we see that we can now rewrite
our equation as 𝑦 equals three times the natural logarithm of two 𝑥 plus
seven. This is great because we know that
the derivative of a constant multiple of an expression in 𝑥 is equal to the
multiple of the derivative of that expression. In other words, the derivative of
three times the natural log of two 𝑥 plus seven is equal to three times the
derivative of the natural log of two 𝑥 plus seven.
But how do we differentiate the
natural log of two 𝑥 plus seven? We’re going to quote the general
result for the derivative of the natural logarithm and we’re going to use the chain
rule. The chain rule says that if 𝑦 is a
function in 𝑢 and 𝑢 itself is a function in 𝑥, then the derivative of 𝑦 with
respect to 𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. And we also know that the
derivative of the natural log of 𝑥 is simply one over 𝑥. So we let 𝑢 be equal to two 𝑥
plus seven such that 𝑦 is equal to the natural log of 𝑢. We know that d𝑢 by d𝑥 is two and
d𝑦 by d𝑢 is one over 𝑢. The derivative then of the natural
log of two 𝑥 plus seven is the product of these. It’s two times one over 𝑢. But we replace 𝑢 with two 𝑥 plus
seven.
And we see that the derivative of
the natural log of two 𝑥 plus seven is two over two 𝑥 plus seven. We recall we said that the
derivative of three times the natural log of two 𝑥 plus seven is three times the
derivative of the natural log of two 𝑥 plus seven. So the derivative is three times
two over two 𝑥 plus seven, which is simply six over two 𝑥 plus seven.
