Find 𝑑𝑦 by 𝑑𝑥, given that 𝑦 equals the natural logarithm of 𝑥 squared plus seven.
𝑦 is the composition of functions. It’s the composition of the natural logarithm function and the function which takes 𝑥 to 𝑥 squared plus seven. And so this is a natural candidate for the chain rule. Let 𝑧 equal 𝑥 squared plus seven. Then, 𝑦 equals the natural logarithm of 𝑧.
Now, the chain rule tells us that 𝑑𝑦 by 𝑑𝑥 is 𝑑𝑦 by 𝑑𝑧 times 𝑑𝑧 by 𝑑𝑥. Let’s apply this rule to our problem. We need to find 𝑑𝑦 by 𝑑𝑧, which is the derivative of the natural logarithm of 𝑧 with respect to 𝑧. And the derivative of the natural logarithm function is the reciprocal function. So 𝑑𝑦 by 𝑑𝑧 is one over 𝑧.
Now, how about 𝑑𝑧 by 𝑑𝑥? Well, 𝑧 is 𝑥 squared plus seven. And differentiating this with respect to 𝑥, we get two 𝑥. We can write this as a fraction as two 𝑥 over 𝑧. But we’re not quite done yet because we have 𝑑𝑦 by 𝑑𝑥 in terms of both 𝑥 and 𝑧 and really would like it in terms of 𝑥 alone if that’s possible. And it is possible. We can substitute 𝑥 squared plus seven for 𝑧. And doing this, we get our final answer two 𝑥 over 𝑥 squared plus seven.
This is a special case of the more general rule that the derivative of the logarithm of a function 𝑓 is the derivative of that function 𝑓 prime divided by the function 𝑓. 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.