Education.com
Try
Brainzy
Try
Plus

Derivative and Integral of Logarithm Help

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

Introduction to Derivative and Integral of Logarithm

We begin by noting these facts:

If a > 0 then

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Math Note : As always, we can state these last formulas more generally as

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

and

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Examples

Example 1

Calculate

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Solution 1

We see that

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

For the second problem, we apply our general formulation with a = 3, u = cos x to obtain

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Similarly,

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Example 2

Integrate

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Solution 2

For clarity we set φ( x ) = cot x, φ′ ( x ) = −csc 2 x . Then our integral becomes

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Resubstituting the expression for φ( x ) now gives that

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

You Try It : Evaluate ∫(log 6 ( x 3 )/ x ) dx .

You Try It : Calculate the integral

x · 3 x 2 d x .

Our new ideas about arbitrary exponents and bases now allow us to formulate a general result about derivatives of powers:

For any real exponent a we have

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Examples

Example 3

Calculate the derivative of Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases .

Solution 3

We have

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

You Try It : Calculate ( d/dx )5 sin x−x 2. Calculate (d/dx) x .

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

Add your own comment