Calculating the Derivative of ln(x)^2

Calculating the Derivative of ln(x)^2
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  • 0:02 Derivative of ln(x)^2
  • 1:05 The Steps to Calculate
  • 4:04 Obtaining the Result
  • 6:13 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we will learn two methods for finding the derivative of the natural logarithm of x-squared. This result is also obtained by using logarithm properties and the definition of the derivative using limits.

Derivative of ln x2

For logarithm functions, we might be conditioned to thinking only positive values of x are allowed. For ln x2, both positive and negative values of x are allowed because x is squared. The only constraint on x is that x ≠ 0.


The function domain includes negative x
The_ln x^2_function


The derivative is the slope of the tangent line at a point. For x > 0, the slope of the tangent line will be positive. For example, at x = 2:


Positive slope for x greater than 0
Positive_slope_for_x_greater_than_0


Likewise, from a graphical observation, tangent lines have a negative slope for x < 0. Consider the tangent line at x = -2:


Negative slope for x less than 0
Negative_slope_for_x_less_than_0


Thus, the derivative of ln x2 must agree with these observations. There are two very straightforward ways to obtain the derivative of ln x2:

  1. Using the chain rule
  2. Using properties of logarithms

Both of these methods use the fact that the derivative of ln x = 1/x.

The Steps to Calculate

Let's look at our first method, the chain rule. How does the chain rule work? Well, first of all, the chain rule is a formula for figuring out the composition of two or more functions. Let's say we have a function with a complicated argument, like sin x2. The function is sine and the argument is x2. If the argument were simply x, we'd differentiate sin x and get cos x.

To use the chain rule, we imagine the function has a simple argument and we write the derivative. In this example, the derivative of sin x2 is cos x2. And then, we multiply by the derivative of the argument. The derivative of x2 is 2x. Thus, using the chain rule, the derivative of sin x2 is cos x2 times 2x or just 2x cos x2.

Step 1: Differentiate with the Chain Rule

The derivative of ln x is 1/x, so the derivative of ln x2 is 1/x2 times the derivative of x2:


using_chain_rule


Then, the derivative of x2 is 2x:


derivative_of_x^2


Step 2: Simplify

1/x2 times 2x can be written as 2x/x2.

Cancelling the common x term:


cancelling_common_x


Now, let's look at our second method, the properties of logarithms, which are basically the properties or characteristics of exponents.

Step 1: Rewrite ln x2 Using Logarithm Properties

The logarithm of x to a power n equals n times the logarithm of x. Thus, ln x2 = 2 ln x.

Step 2: Differentiate


derivative_of_2ln_x^2


The constant 2 comes out of the differentiation:


pulling_out_the_2


Leaving us with the derivative of ln x, which is 1/x:


derivative_of_ln_x


Step 3: Simplify

The 2 multiplied by 1/x is written as 2/x:


simplifying


Thus, the derivative of ln x2 is 2/x. Note this result agrees with the plots of tangent lines for both positive and negative x. For x = 2, the derivative is 2/2 = 1, which agrees with the plot. And for x = -2, the derivative is 2/(-2) = -1, which agrees with the negative sloping tangent line at x = -2.

Obtaining The Result

Using the limits definition of the derivative, a nice definition for the exponential along with logarithm properties, we can differentiate ln x2.

First, the definition of the derivative:


definition_of_derivative


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