### Video Transcript

If 𝑓 of 𝑥 is equal to three ln

multiplied by two 𝑥 plus four ln 𝑥, find the derivative of 𝑓 of 𝑥 when 𝑥 is

equal to one.

Now, the first thing we can do if

we’re looking to differentiate our function is take our constant term out because

this wouldn’t affect the differentiation. So we’ve now got three multiplied

by and then the derivative of ln multiplied by two 𝑥 plus four ln 𝑥. Now in order to solve the problem

and work out the derivative of the expression, we’re gonna have to use one of our

derivative rules. And this is one that’s based around

our natural logarithm, which is ln.

So we know that if we want to find

the derivative of ln — and in this case, we’re gonna say 𝑢 of 𝑥, so just a

function — then it’s equal to one over 𝑢 of 𝑥 multiplied by the derivative of 𝑢

of 𝑥. So therefore, what we’re gonna get

is three multiplied by and then we’ve got one over two 𝑥 plus four ln 𝑥. And that’s because that was our 𝑢

of 𝑥. So that was our function when we’re

looking at the rule for differentiating something like this. Then, we multiply this by the

derivative of two 𝑥 plus four ln 𝑥.

And as we know with

differentiation, what we can do is when we’re differentiating an expression like

this is we can differentiate each term separately. So we can differentiate two 𝑥 and

we can differentiate four ln 𝑥. So first of all, if we

differentiate two 𝑥, we’re just gonna get two. And that’s because what we do is we

multiply the coefficient by the exponent. So we’d have two multiplied by one

which is just two. And then, we reduce exponent from

one to zero. So we get two multiplied by one

which is just two.

And then, what we’re gonna do to be

able to differentiate the second term is use another one of our rules. And that is if we have derivative

of ln 𝑥, it’s just equal to one over 𝑥. So if we’re looking to

differentiate for ln 𝑥, the first thing we can do — as normal — is take the four,

the constant, out. So we have four multiplied by the

derivative of ln 𝑥, which is just gonna give us four over 𝑥. And that’s because we’re gonna have

four multiplied by one over 𝑥 which is four over 𝑥. So what this does is that it leaves

us with an expression. And this expression is three

multiplied by two because that was the derivative of two 𝑥 plus four over 𝑥

because that was the derivative of four ln 𝑥. And this is all over two 𝑥 plus

four ln 𝑥.

Now, at this stage, we’d think

about maybe simplifying. But there’s no need because what

we’re trying to find is the value of this expression, so the value of the

derivative, when 𝑥 is equal to one. And therefore, to do this, what we

need to do is we need to substitute in one for 𝑥 at every point in our derivative

expression. So when we do that, we get three

multiplied by two plus four over one over two multiplied by one plus four ln

one. So then, what we’re gonna get is 18

over two. And that’s because we got three

multiplied by two plus four. Well, two plus four is six. Three sixes are 18. Then I’ll need denominator: we’ve

got two multiplied by one which is two then add four ln one, well ln one is just

equal to zero. So we’re left with two on the

denominator. So therefore, this is gonna give us

a final answer of nine.

So therefore, we can say that if

function 𝑓 of 𝑥 is equal to three ln two 𝑥 plus four ln 𝑥, then the first

derivative value when 𝑥 is equal to one is going to be nine.