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Derivative of Logarithm Function Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

The Logarithm Function and the Derivative

Now you will see why our new definition of logarithm is so convenient. If we want to differentiate the logarithm function we can apply the Fundamental Theorem of Calculus:

Transcendental Functions 6.1 Logarithm Basics

More generally,

Transcendental Functions 6.1 Logarithm Basics

Example 1

Calculate

Transcendental Functions 6.1 Logarithm Basics

Solution 1

For the first problem, we let u = 4 + x and du/dx = 1. Therefore we have

Transcendental Functions 6.1 Logarithm Basics

Similarly,

Transcendental Functions 6.1 Logarithm Basics

You Try It : What is the derivative of the function ln( x3 + x2 )?

Now we examine the graph of y = ln x. Since

Transcendental Functions 6.1 Logarithm Basics

we know that ln x is an increasing, concave down function whose graph passes through (1,0). There are no relative maxima or minima (since the derivative is never 0). Certainly ln .9 < 0; the formula ln(.9 n ) = n ln .9 therefore tells us that ln x is negative without bound as x → 0 + . Since ln x = − ln(1/ x ), we may also conclude that ln x is positive without bound as x → +∞. A sketch of the graph of y = ln x appears in Fig. 6.2.

We learned in the last paragraph that the function ln x takes negative values which are arbitrarily large in absolute value when x is small and positive. In particular, the negative y axis is a vertical asymptote. Since ln(1/ x ) = − ln x , we then find that ln x takes arbitrarily large positive values when x is large and positive. The graph exhibits these features.

Since we have only defined the function ln x when x > 0, the graph is only sketched in Fig. 6.2 to the right of the y -axis. However it certainly makes sense to discuss the function ln | x | when x ≠ 0 (Fig. 6.3):

Transcendental Functions 6.1 Logarithm Basics

Fig. 6.2

Transcendental Functions 6.1 Logarithm Basics

Fig. 6.3

If x ≠ 0 then

Transcendental Functions 6.1 Logarithm Basics

In other words,

Transcendental Functions 6.1 Logarithm Basics

More generally, we have

Transcendental Functions 6.1 Logarithm Basics

and

Transcendental Functions 6.1 Logarithm Basics

Example 2

Calculate

Transcendental Functions 6.1 Logarithm Basics

Solution 2

Transcendental Functions 6.1 Logarithm Basics

You Try It : Calculate the integral

Transcendental Functions 6.1 Logarithm Basics

You Try It : Calculate the integral

Transcendental Functions 6.1 Logarithm Basics

Example 3

Evaluate the integral

Transcendental Functions 6.1 Logarithm Basics

Solution 3

For clarity we set φ( x ) = 3 sin x −4, φ′( x ) = 3(cos x ). The integral then has the form

Transcendental Functions 6.1 Logarithm Basics

Resubstituting the expression for φ( x ) yields that

Transcendental Functions 6.1 Logarithm Basics

You Try It : Evaluate Transcendental Functions 6.1 Logarithm Basics.

Example 4

Calculate

∫ cot x dx.

Solution 4

We rewrite the integral as

Transcendental Functions 6.1 Logarithm Basics

For clarity we take φ(x) = sin x , φ′( x ) = cos x. Then the integral becomes

Transcendental Functions 6.1 Logarithm Basics

Resubstituting the expression for φ yields the solution:

∫ cot x dx = ln |sin x| + C.

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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