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

Describe order of operations with example, Describe Order of Operations wit...

Describe Order of Operations with example? The order of operations is a set of rules that describe the order in which math operations are done. Try doing this math problem:

Lines- common polar coordinate graphs, Lines- Common Polar Coordinate Graph...

Lines- Common Polar Coordinate Graphs A few lines have quite simple equations in polar coordinates. 1.  θ = β We are able to see that this is a line by converting to Car

Algebra, sir i want to ask u a question and that is if we simplify this wha...

sir i want to ask u a question and that is if we simplify this what will be the answer.(9x-45z+6y-100z+5x)

What is the probability that the card is a queen, Five cards - the ten, jac...

Five cards - the ten, jack, queen, king and ace, are well shuffled with their face downwards. One card is then picked up at random. (i)  What is the probability that the card is

Multiples, The sum of the smallest and largest multiples of 8 up to 60 is?

The sum of the smallest and largest multiples of 8 up to 60 is?

what is probability that point will be chosen from triagle, In the adjoini...

In the adjoining figure ABCD is a square with sides of length 6 units points P & Q are the mid points of the sides BC & CD respectively. If a point is selected at random from the i

Distance traveled, a) Determine the distance traveled among t = 0 and  t =∏...

a) Determine the distance traveled among t = 0 and  t =∏/2 by a particle P(x, y) whose position at time t is given by Also check your result geometrically.  (5) b) D

The shortest distance between the line y-x=1 and curve x=y^2, Any point on ...

Any point on parabola, (k 2 ,k) Perpendicular distance formula: D=(k-k 2 -1)/2 1/2 Differentiating and putting =0 1-2k=0 k=1/2 Therefore the point is (1/4, 1/2) D=3/(32 1/2

Segmentation, what is segmentation and how to used as per the market with e...

what is segmentation and how to used as per the market with example?

Right-handed limit, Right-handed limit We say provided we can m...

Right-handed limit We say provided we can make f(x) as close to L as we desire for all x sufficiently close to a and x>a without in fact letting x be a.

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