Video: Differentiating Logarithmic Functions Using the Chain Rule

Find 𝑑𝑦/𝑑π‘₯, given that 𝑦 = ln(π‘₯Β² + 7).

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Video Transcript

Find 𝑑𝑦 by 𝑑π‘₯, given that 𝑦 equals the natural logarithm of π‘₯ squared plus seven.

𝑦 is the composition of functions. It’s 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 𝑧 equal π‘₯ squared plus seven. Then, 𝑦 equals the natural logarithm of 𝑧.

Now, the chain rule tells us that 𝑑𝑦 by 𝑑π‘₯ is 𝑑𝑦 by 𝑑𝑧 times 𝑑𝑧 by 𝑑π‘₯. 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 𝑑π‘₯? Well, 𝑧 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 𝑧. But we’re not quite done yet because we have 𝑑𝑦 by 𝑑π‘₯ in terms of both π‘₯ and 𝑧 and really would like it in terms of π‘₯ alone if that’s possible. And it is possible. We can substitute π‘₯ squared plus seven for 𝑧. 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 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.

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