4.6.2 The Derivative of the Natural Logarithm

We have formulas for the derivatives of all but one of the functions we have encountered so far — all except the natural logarithm. Before we find this formula, let's return to the definition of the derivative and look at difference quotients for `ln x`.

In Figure 2 we graph the difference quotient function

which should closely approximate the derivative of `ln x`. Do you recognize this function?

Activity 1

1. Use the graph to estimate the difference quotient for `x=1text[/]4`, `x=1text[/]2,` `x=1`, `x=2`, and `x=4`.

2. On the basis of your estimates in (a), make a conjecture about the derivative of `ln x`.

Comment on Activity 1

Now let's find the formula for the derivative of `ln x` (and, we hope, verify your conjecture). Implicit differentiation turns out to be just what we need. If `y=ln x`, then, by the definition of the natural logarithm, `e^y=x.` If we differentiate both sides of this equation with respect to `x`, we obtain

which leads to

But, since `e^y=x`, we find

We have shown that

Since the natural logarithm is defined only for positive numbers `x`, this formula is correct for all `x>0`.

Checkpoint 3

Implicit differentiation is a useful way to obtain differentiation formulas. In Example 1 we use this method to derive again the formula we know already for the derivative of the square root function.

Example 1

Apply the implicit differentiation technique to the equation

to obtain the formula

Solution   Differentiating both sides of the equation `y^2=x` with respect to `x`, we find

Solving for `dytext[/]dx`, we get

Since `y=sqrt(x)`, we may substitute to obtain