Derivatives for logarithm, Mathematics

Assignment Help:

Logarithm Functions : Now let's briefly get the derivatives for logarithms.  In this case we will have to start with the following fact regarding functions that are inverses of each other.

Fact 2 : If f(x) & g(x) are inverses of each other then,

                                                               g′ ( x ) = 1/ f ′ ( g ( x ))

Hence, how is this issue useful to us? Well recall that the natural exponential function and the natural logarithm function are inverses of each other and we know derivative of the natural exponential function.

Hence, if we have f ( x ) = ex  and g ( x ) =ln x then,

g′ ( x ) = 1/f ′ ( g ( x )) = 1/ e g ( x )  = 1/ e ln x  =1/ x

The final step just utilizes the fact that the two functions are inverses of each other.

Putting this all together gives,

                              d (ln x )/dx = 1/x                   x>0

Note as well that we have to require that x > 0 as it is required for the logarithm and hence have to also be needed for its derivative.  It can also be illustrated that,

            d(ln  |x| ) /dx= 1/x                                x ≠ 0

By using this all we have to avoid is x=0

In this, unlike the exponential function case, actually we can determine the derivative of the general logarithm function. All that we required is the derivative of the natural logarithm, that we only found, and the change of base formula.  By using the change of base formula we may write a general logarithm as,

                                               loga x = ln x /ln a

Differentiation is then fairly simple.

d(log a x)/dx = d(lnx/lna)/dx

                      = (1/lna )(d(lnx)/dx

                     = 1/xlna

We took benefit of the fact that a was a constant and thus ln a is also a constant and can be factored of the derivative.  Putting all of this together gives,

                                               d (logax)/dx =1/(xlna)

Following is a summary of the derivatives in this section.

d (ex )/dx= ex                                           d (a x ) / dx = a x ln a

d (ln x ) /dx= 1                                          d(log a x)/dx  = 1/xlna


Related Discussions:- Derivatives for logarithm

Ratio, how to do them?

how to do them?

Solve the inequality |x - 1| + |x - 2|, Solve the inequality |x - 1| + |x -...

Solve the inequality |x - 1| + |x - 2|≤ 3. Working Rule:    First of all measure the expression to zero whose modulus happens in the given inequation and from this search the va

What is his test average, Steve earned a 96 percent on his ?rst math test, ...

Steve earned a 96 percent on his ?rst math test, a 74% on his second test, and an 85 percent on his third test. What is his test average? Add the test grades (96 + 74 + 85 = 25

Write the next two terms, Write the next two terms √12, √27, √48, √75.........

Write the next two terms √12, √27, √48, √75................... Ans:    next two terms √108 , √147 AP is 2 √3 , 3 √3 , 4 √3 , 5 √3 , 6 √3 , 7 √3 ......

Correlation and regression, Correlation and Regression Correlation ...

Correlation and Regression Correlation CORRELATION is an important statistical concept which refers to association or interrelationship among variables. The reasons of

Determine the angle, In parallelogram ABCD, m∠A = 3x + 10 and m∠D = 2x + 30...

In parallelogram ABCD, m∠A = 3x + 10 and m∠D = 2x + 30, Determine the m∠A. a. 70° b. 40° c. 86° d. 94° d. Adjacent angles in a parallelogram are supplementary. ∠A a

Find out if the sets of vectors are parallel or not, Determine or find out ...

Determine or find out if the sets of vectors are parallel or not. (a) a → = (2,-4,1), b = (-6, 12 , -3) (b) a → = (4,10), b = (2,9) Solution (a) These two vectors

SIMPLE INTEREST, A payday loan company charges a $95 fee for a $500 payday ...

A payday loan company charges a $95 fee for a $500 payday loan that will be repaid in 11 days. Treating the fee as interest paid, what is the equivalent annual interest rate?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd