# Question Video: Deriving the Inverse Function of an Exponential Function Mathematics

Find dπ¦/dπ₯, given that π¦ = ln (2π₯ + 7)Β³.

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### Video Transcript

Find dπ¦ by dπ₯, given that π¦ is equal to the natural log of two π₯ 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 weβre working with the natural logarithm, itβ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 π₯ to the π¦th power is π¦ times the natural logarithm of π₯. And we see that we can now rewrite our equation as π¦ equals three times the natural logarithm of two π₯ plus seven. This is great because we know that the derivative of a constant multiple of an expression in π₯ is equal to the multiple of the derivative of that expression. In other words, the derivative of three times the natural log of two π₯ plus seven is equal to three times the derivative of the natural log of two π₯ plus seven.

But how do we differentiate the natural log of two π₯ plus seven? Weβre going to quote the general result for the derivative of the natural logarithm and weβre going to use the chain rule. The chain rule says that if π¦ is a function in π’ and π’ itself is a function in π₯, then the derivative of π¦ with respect to π₯ is equal to dπ¦ by dπ’ times dπ’ by dπ₯. And we also know that the derivative of the natural log of π₯ is simply one over π₯. So we let π’ be equal to two π₯ plus seven such that π¦ is equal to the natural log of π’. We know that dπ’ by dπ₯ is two and dπ¦ by dπ’ is one over π’. The derivative then of the natural log of two π₯ plus seven is the product of these. Itβs two times one over π’. But we replace π’ with two π₯ plus seven.

And we see that the derivative of the natural log of two π₯ plus seven is two over two π₯ plus seven. We recall we said that the derivative of three times the natural log of two π₯ plus seven is three times the derivative of the natural log of two π₯ plus seven. So the derivative is three times two over two π₯ plus seven, which is simply six over two π₯ plus seven.