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)).
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.
We know the derivative of ln(√x) is 1/(2x). Putting all this information together, we get the following:h ‘ (x) = 1/(2x) + 0 = 1/(2x)We now have a formula that will give us the rate at which Mandi was growing x months after the growth spurt started.For example, if she wants to know what her growth rate was four months in, she simply needs to plug x = 4 into the derivative formula.1 / (2x) = 1 / (2;4) = 1/8Four months after Mandi’s growth spurt started, she was growing at a rate of 1/8 of an inch each month. Similarly, if she wants to know what her growth rate was 30 months in, she would plug x = 30 into the derivative formula.1/ (2x) = 1 / (2⋅30) = 1/60Here Mandi was growing at a rate of 1/60 of an inch per month at 30 months in.
So she was right about the fact that she grew more quickly at the beginning of her growth spurt.Growth is an ordinary human process that we all experience and we can use the derivative of ln(√x) to analyze it.