### Video Transcript

Find ππ¦ by ππ₯, given that π¦ equals the natural logarithm of π₯ squared plus seven.

π¦ is the composition of functions. Itβs the composition of the natural logarithm function and the function which takes π₯ to π₯ squared plus seven. And so this is a natural candidate for the chain rule. Let π§ equal π₯ squared plus seven. Then, π¦ equals the natural logarithm of π§.

Now, the chain rule tells us that ππ¦ by ππ₯ is ππ¦ by ππ§ times ππ§ by ππ₯. Letβs apply this rule to our problem. We need to find ππ¦ by ππ§, which is the derivative of the natural logarithm of π§ with respect to π§. And the derivative of the natural logarithm function is the reciprocal function. So ππ¦ by ππ§ is one over π§.

Now, how about ππ§ by ππ₯? Well, π§ is π₯ squared plus seven. And differentiating this with respect to π₯, we get two π₯. We can write this as a fraction as two π₯ over π§. But weβre not quite done yet because we have ππ¦ by ππ₯ in terms of both π₯ and π§ and really would like it in terms of π₯ alone if thatβs possible. And it is possible. We can substitute π₯ squared plus seven for π§. And doing this, we get our final answer two π₯ over π₯ squared plus seven.

This is a special case of the more general rule that the derivative of the logarithm of a function π is the derivative of that function π prime divided by the function π. This is a very useful rule and it pops up now and again. And it is itself a special case of the more general chain rule.