Solving the Derivative of ln(sqrt x)

Solving the Derivative of ln(sqrt x)
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  • 0:04 Steps to Solve
  • 2:40 Application
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this video, we'll learn how to find the derivative of ln(sqrt(x)) and review the chain rule for derivatives. After we've found the derivative, we'll see how it can be applied to an everyday situation.

Steps to Solve

If we want to find the derivative of ln(√x), we will have to make use of the chain rule for derivatives. The chain rule for derivatives is a rule that we use to find the derivative of functions of the form f(g(x)).


derlnsqrx1


Notice that if we let f(x) = ln(x) and g(x) = √x, then f(g(x)) = ln(√x), which tells us that we can use the chain rule to find the derivative of ln(√x).

Let's think about what else we'll need to know to find this derivative. The chain rule states that if h(x) = f(g(x)), then h ' (x) = f ' (g(x)) ⋅ g ' (x)

In our example, f(x) = ln(x) and g(x) = √x. To use the chain rule, we're going to have to find f ' (g(x)), and g ' (x). Both of these derivatives are well-known.

  • The derivative of ln(x) is 1 / x.
  • The derivative of √x is (1/2)x(-1/2), or 1/(2√x).

These facts will be helpful in our quest for the derivative. Since the derivative of ln(x) is 1/x, we have that f ' (x) = 1/x, so f ' (g(x)) = 1/(√x). Also, we know that g ' (x) = 1 / (2√x). So all we have to do is plug these into the chain rule formula and simplify to get our derivative.


derlnsqrx2

Application

Finding the derivative of ln(√x) wasn't so hard. Now, let's consider a scenario where being able to do so might come in handy. Suppose a girl named Mandi has gone through a growth spurt over the past few years. We can model her growth over these 36 months by using the function h(x) = ln(√x) + 1, where h(x) is the number of inches that she's grown during this growth spurt, and x is the number of months since the growth spurt started.

Mandi notices that when her growth spurt first began, she grew very quickly, but it seems to have slowed, though she's still growing. We can illustrate this trend by graphing the function.


derlnsqrtx4


This gets Mandi to thinking about just how quickly she was growing at different times over the past 36 months. We can use the derivative of ln(√x) to answer her question.

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