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

Logarithems , y=x4/4lnx-x4/16 then dy/dx=? Solution) dy/dx=-x^3/4(2/lnx-...

y=x4/4lnx-x4/16 then dy/dx=? Solution) dy/dx=-x^3/4(2/lnx-1)^2.    ^ means power

Trignometric function, If tanx+secx=sqr rt 3, 0 Ans) sec 2 x=(√3-tanx) 2...

If tanx+secx=sqr rt 3, 0 Ans) sec 2 x=(√3-tanx) 2 1+tan 2 x=3+tan 2 x-2√3tanx 2√3tanx=2 tanx=1/√3 x=30degree

Dimensions and degree of an expression, Binomials, Trinomials and P...

Binomials, Trinomials and Polynomials which we have seen above are not the only type. We can have them in a single variable say 'x' and of the form x 2 + 4

Course work2 , (b) The arity of an operator in propositional logic is the n...

(b) The arity of an operator in propositional logic is the number of propositional variables that it acts on – for example, binary operations (e.g, AND, OR, XOR…) act on two propo

Real exponents, It is a fairly short section.  It's real purpose is to ackn...

It is a fairly short section.  It's real purpose is to acknowledge that the exponent properties work for any exponent.  We've already used them on integer and rational exponents al

What are factor trees explain, What are Factor Trees explain? In algebr...

What are Factor Trees explain? In algebra, we often need to factor a number into its prime factors. One way to do this is to use a factor tree. This is a network of numbers, st

First order linear differential equation, Newton's Second Law of motion, wh...

Newton's Second Law of motion, which recall from the earlier section that can be written as: m(dv/dt) = F (t,v) Here F(t,v) is the sum of forces which act on the object and m

Find the lesser of two consecutive positive even integers, Find the lesser ...

Find the lesser of two consecutive positive even integers whose product is 168. Let x = the lesser even integer and let x + 2 = the greater even integer. Because product is a k

Mensuration, How do mensuration relate to the real life issues

How do mensuration relate to the real life issues

Sketch the parametric curve for parametric equations, Sketch (draw) the par...

Sketch (draw) the parametric curve for the subsequent set of parametric equations. x = t 2 + t y = 2t -1 Solution At this point our simply option for sketching a par

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