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Chapter Contents

• Differentiation of Transcendental Functions
• 1. Derivatives of Sin, Cos and Tan Functions
• 2. Derivatives of Csc, Sec and Cot Functions
• 3. Derivatives of Inverse Trigonometric Functions
• 4. Applications: Derivatives of Trigonometric Functions
• 5. Derivative of the Logarithmic Function
• 6. Derivative of the Exponential Function
• 7. Applications: Derivatives of Logarithmic and Exponential Functions

• Comments, Questions?

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From the math blog...

5. Derivative of the Logarithmic Function

by M. Bourne

First, let's look at a graph of the log function with base e, that is f(x) = log_e(x) (usually written "ln x"). The tangent at x = 2 is included on the graph.

The slope of that tangent of y = ln x at x = 2 is 1/2. (We can observe this from the graph, by looking at the ratio rise/run).

Explore the slope of log x

In the following interactive graph, the blue curve is y = ln x and the magenta line is the tangent at a point on the curve.

You can drag the dot representing the point (drag in a left-right motion), and observe the changing slope.

For each value of x, what is the slope? (Pause at x = 1, x = 2, x = 3, etc and observe the slope at those points.)

From the above exercise, hopefully you found the following slopes for various values of x.

If y = ln x,

x	1	2	3	4	5
slope of graph	1	1/2	1/3	1/4	1/5
1/x	1	1/2	1/3	1/4	1/5

We see that the slope of the graph for each value of x is equal to 1/ x. This works for any positive value of x (we cannot have the logarithm of a negative number, of course).

If we did many more examples, we could conclude that the derivative of the logarithm function y = ln x is

dy/dx = 1/x.

Note 1: Actually, this result comes from first principles.

Note 2: We are using logarithms with base e. If you need a reminder about log functions, check out Log base e from before.

Derivative of the Logarithm Function y = ln x

The derivative of the logarithmic function y = ln x is given by:

You will see it written in a few other ways as well. The following are equivalent:

If y = ln x, then

For some problems, we can use the logarithm laws to simplify our log expression before differentiating it.

Example 1

Find the derivative of

y = ln 2x

Answer

Example 2

Find the derivative of

y = ln x²

Answer

Derivative of y = ln u (where u is a function of x)

Unfortunately, we can only use the logarithm laws to help us in a limited number of logarithm differentiation question types.

Most often, we need to find the derivative of a logarithm of some function of x. For example, we may need to find the derivative of y = 2 ln (3x² − 1).

We need the following formula to solve such problems.

If

y = ln u

and u is some function of x, then:

where u' is the derivative of u

Another way to write this is

You might also see the following form. It means the same thing.

If

y = ln f(x),

then the derivative of y is given by:

Example 3

Find the derivative of

y = 2 ln (3x² − 1).

Answer

Example 4

Find the derivative of

y = ln(1 − 2x)³.

Here it is in LiveMath:

LIVEMath

Normal answer:

Answer

Example 5

Find the derivative of

Answer

Differentiating Logarithmic Functions with Bases other than e

If

u = f(x) is a function of x,

and

y = log_b u is a logarithm with base b,

then we can obtain the derivative of the logarithm function with base b using:

where

u' is the derivative of u

log_be is a constant. See change of base rule to see how to work out such constants on your calculator.)

Note 1: This formula is derived from first principles.

Note 2: If we choose e as the base, then the derivative of ln u, where u is a function of x, simply gives us our formula above:

[Recall that log_ee = 1.]

[See the chapter on Exponential and Logarithmic Functions base e if you need a refresher on all this.]

Example 6

Find the derivative of y = log₂6x.

Answer

Example 7

Find the derivative of y = 3 log₇(x² + 1).

Answer

Note: Where possible, always use the properties of logarithms to simplify the process of obtaining the derivatives.

Exercises

1. Find the derivative of 

y = ln(2x³ −x)².

Answer

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2. Find the derivative of 

y = ln(cos x²).

Answer

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3. Find the derivative of 

y = x ln³ x.

Answer

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4. Find the derivative of

3 ln xy + sin y = x².

Answer

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5. Find the derivative of

y = (sin x)^x

by first taking logarithms of each side of the equation.

Answer