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Author Topic: Differentiating exponentials and logarithms  (Read 3162 times)

nid404

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Differentiating exponentials and logarithms
« on: July 23, 2010, 03:58:25 pm »
One of our members asked for help, so I thought of posting it up for the others to benefit as well :)

\frac{d(u.v)}{dx}= v\frac{du}{dx}+ u\frac{dv}{dx}

\frac{d(u.v.w)}{dx}= vw\frac{du}{dx}+ uw\frac{dv}{dx}+uv\frac{dw}{dx}


\frac{d(u/v)}{dx}= \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}


If y= b^x

same goes for derivative of  ln x

exponentials

y= e^x

log y= xlog e




log y= x log b

\frac{d(logx)}{dx}= \frac{1}{x}\frac{d(x)}{dx}

log y= x log b

\frac{1}{y}\frac{d(y)}{dx}= x\frac{d(logb)}{dx}+logb\frac{d(x)}{dx}

\frac{1}{y}\frac{d(y)}{dx}= x(0) + log b (1)

\frac{1}{y}\frac{d(y)}{dx}= log b\frac{d(x)}{dx}---> in this case is unit

\frac{d(y)}{dx}=y log b

since y=b^x

\frac{d(y)}{dx}= b^x log b \frac{d(x)}{dx}


Same goes for derivative of ln x

y=e^x

ln y= x ln e


\frac{1}{y}\frac{d(y)}{dx}= ln e\frac{d(x)}{dx}+ x\frac{lne}{dx}

\frac{1}{y}\frac{d(y)}{dx} = 1 + 0

\frac{d(y)}{dx}= 1y

y=e^x

\frac{d(y)}{dx}=1e^x= e^x



lol...that sure took looong to type....I will solve some sample problems from my textbook--Pure mathematics 2&3 Hugh Neill and Douglas Quadling...in a while  ;)
« Last Edit: July 23, 2010, 04:52:15 pm by ~Ahana~ »
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #1 on: July 23, 2010, 05:06:59 pm »
Ex 4 A of the textbook

Differentiate

I'm writing the numbers as per the textbook so the ones who use it find it easier to look for.

1)

a) e3x
\frac{dy}{dx}= e3x\frac{d(3x)}{dx}

\frac{dy}{dx} =e3xX3= 3e3x

h) 3e. e2+4x

=3e3+4x

\frac{dy}{dx}= 3e3+4x\frac{d(3+4x)}{dx}
  
\frac{dy}{dx}=  3e3+4xX 4

\frac{dy}{dx}= 12e3+4x


« Last Edit: July 23, 2010, 05:13:00 pm by ~Ahana~ »
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #2 on: July 23, 2010, 05:20:58 pm »
4)
c) y= e^\sqrt(1-x^2)
    y= e^\sqrt(1-x^2) X \frac{d\sqrt(1-x^2)}{dx}
    y=  e^\sqrt(1-x^2) X 0.5 (1-x2)-1/2 X 2x
    y= e^\sqrt(1-x^2) - x (1-x2)-1/2
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Offline Saladin

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Re: Differentiating exponentials and logarithms
« Reply #3 on: July 23, 2010, 08:14:27 pm »
+ REP. Good job.
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #4 on: July 24, 2010, 07:03:04 am »
hey thanks

Exercise 4 B

1)
j) dy/dx ln(x(x+1))

dy/dx ln (x2+x)

1/(x2+x) X d (x2+x)/dx

(2x+1)/ (x2+x)

4) b)

y= 0.5 ln (2+ x4)
  = 1/2 X 1/(2+ x4) X d(2+ x4)/dx
  =1/2 X 1/(2+ x4) X 4x3
  =  4x3/ 4+2x4
  =2x3/2+x4
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #5 on: July 24, 2010, 07:43:12 am »
If you have any further doubts, you can post them here  :)
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Offline mousa

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Re: Differentiating exponentials and logarithms
« Reply #6 on: July 24, 2010, 07:57:17 am »
Your awesome nidd!!.. Thank you  :D
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Offline mousa

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Re: Differentiating exponentials and logarithms
« Reply #7 on: July 24, 2010, 07:58:28 am »
+ REP!!!!!!!!!!!!!!!  ;D
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Offline mousa

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Re: Differentiating exponentials and logarithms
« Reply #8 on: July 24, 2010, 08:04:54 am »
in the same chapter, but the section regarding reciprocal integrals, can you do a few well explained examples???

Thanks.
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #9 on: July 24, 2010, 08:36:12 am »
yup sure :)

\int\frac{1}{x} dx = ln x + k if x>0

\int\frac{1}{x} dx = ln -x = k if x<0

Exercise  4 D

1) e) \int^4_2\frac{1}{-x-1}dx

        \int^4_2ln(-x-1)  
        ln(-4-1)- ln (-2-1)
        ln(-5)-ln (-3)
       \frac{ln(-5)}{-3}= \frac{-ln5}{3}
« Last Edit: July 24, 2010, 08:38:20 am by ~Ahana~ »
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #10 on: July 24, 2010, 08:39:04 am »
I'll do a few more once I get back  :)
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #11 on: July 25, 2010, 08:27:23 am »
4D

2) y= ln |2x-3| 

when x= -2, calculate the value of y and find dy/dx

y= ln|2(-2)-3|
 = ln 7


dy/dx= 1/ 2x-3 d(2x-3)/dx

dy/dx= 2/2x-3

dy/dx= 2/ -7
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Offline mousa

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Re: Differentiating exponentials and logarithms
« Reply #12 on: July 25, 2010, 12:16:57 pm »
I would realy appreciate if you can do a few complicated ones (integrals).

Thanks.
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nid404

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Re: Differentiating exponentials and logarithms
« Reply #13 on: July 25, 2010, 12:22:37 pm »
You can tell me the ones you want me to do...I'll do 'em  :)
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Offline mousa

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Re: Differentiating exponentials and logarithms
« Reply #14 on: July 25, 2010, 01:06:02 pm »
Quote from: ~Ahana~ on July 25, 2010, 12:22:37 pm
You can tell me the ones you want me to do...I'll do 'em  :)

ok. Ex.4D 1,F

Miscellaneous. 1 E., Q4,5,6,7


I guess those should be enough.

Thnaks in advance.  :)
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