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

Rotational symmetry .., write down the order of rotational symmetry of the ...

write down the order of rotational symmetry of the rectangle

Vector function - three dimensional spaces, Vector Function The good wa...

Vector Function The good way to get an idea of what a vector function is and what its graph act like is to look at an instance.  Thus, consider the following vector function.

Evaluate inverse tangents , Evaluate following limits. Solution ...

Evaluate following limits. Solution Here the first two parts are actually just the basic limits including inverse tangents and can easily be found by verifying the fol

Word problems, if I read 6 hours of spring break how many minutes did ...

if I read 6 hours of spring break how many minutes did read

Play and learn maths, PLAY AND LEARN :  Children can learn many basic math...

PLAY AND LEARN :  Children can learn many basic mathematical concepts through games. They enjoy Mathematical concepts can be playing within familiar contexts. Their games also gen

Study market, what toold we need to study market

what toold we need to study market

Trigonometric ratios, to difine trigonometric ratios of an angle,is it nece...

to difine trigonometric ratios of an angle,is it necessary that the initial ray of the angle must be positive x-axis?

Sum of their areas is given find radii of the two circles, Two circles touc...

Two circles touch externally. The sum of their areas is 58 π cm 2 and the distance between their centres is 10 cm. Find the radii of the two circles. (Ans:7cm, 3cm) Ans:

The sum of two integers is 36 what is the smaller number, The sum of two in...

The sum of two integers is 36, and the difference is 6. What is the smaller of the two numbers? Let x = the ?rst integer and let y = the second integer. The equation for the su

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