IGCSE/GCSE/O & A Level/IB/University Student Forum

Qualification => Reference Material => GCE AS & A2 Level => Revison Notes => Topic started by: nid404 on July 23, 2010, 03:58:25 pm

Title: Differentiating exponentials and logarithms
Post by: nid404 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 ;)
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 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) e^3x
$\frac{dy}{dx}$ = e^3x $\frac{d(3x)}{dx}$

$\frac{dy}{dx}$ =e^3xX3= 3e^3x

h) 3e. e^2+4x

=3e^3+4x

$\frac{dy}{dx}$ = 3e^3+4x $\frac{d(3+4x)}{dx}$

$\frac{dy}{dx}$ = 3e^3+4xX 4

$\frac{dy}{dx}$ = 12e^3+4x
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 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
Title: Re: Differentiating exponentials and logarithms
Post by: Saladin on July 23, 2010, 08:14:27 pm: + REP. Good job.
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 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
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 24, 2010, 07:43:12 am: If you have any further doubts, you can post them here :)
Title: Re: Differentiating exponentials and logarithms
Post by: mousa on July 24, 2010, 07:57:17 am: Your awesome nidd!!.. Thank you :D
Title: Re: Differentiating exponentials and logarithms
Post by: mousa on July 24, 2010, 07:58:28 am: + REP!!!!!!!!!!!!!!! ;D
Title: Re: Differentiating exponentials and logarithms
Post by: mousa 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.
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 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) $\int^4_2$ $\frac{1}{-x-1}dx$

   $\int^4_2$ $ln(-x-1)$
   $ln(-4-1)- ln (-2-1)$
   $ln(-5)-ln (-3)$
   $\frac{ln(-5)}{-3}$ = $\frac{-ln5}{3}$
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 24, 2010, 08:39:04 am: I'll do a few more once I get back :)
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 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
Title: Re: Differentiating exponentials and logarithms
Post by: mousa on July 25, 2010, 12:16:57 pm: I would realy appreciate if you can do a few complicated ones (integrals).

Thanks.
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 25, 2010, 12:22:37 pm: You can tell me the ones you want me to do...I'll do 'em :)
Title: Re: Differentiating exponentials and logarithms
Post by: mousa 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. :)
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 25, 2010, 01:22:57 pm: ohkay gimme some time.
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 26, 2010, 12:34:51 pm: 4 D

1 f) $\int^0_-1 2+$ $\frac{1}{x-1} dx$

$\int^0_-1$ $2x+ln |(x-1)|$

= 0- (-2+ ln|(-2)|
= 2-ln 2
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 26, 2010, 12:38:37 pm: I'll have a nap and do the rest ;D
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 26, 2010, 03:02:46 pm: Quote from: mousa on July 25, 2010, 01:06:02 pm

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

1) e) $\frac{ln 2-x}{3-x}$

ln |2-x| - ln |3-x|
dy/dx= 1/2-x d(2-x)dx - 1/3-x d(3-x)/dx
= -1/2-x + 1/3-x
Title: Re: Differentiating exponentials and logarithms
Post by: nid404 on July 26, 2010, 03:45:13 pm: 5) $\int(2+e^-x) dx$

$2x+e^-x X d(-x)/dx + k$
$2x-e^(-x) + k$

6) y= 2x²-lnx

dy/dx= 4x- 1/x

4x-1/x= 0
4x²=1
x²=0.25
x= 0.5

dy/dx greater than zero...hence minimum
Title: Re: Differentiating exponentials and logarithms
Post by: mousa on July 27, 2010, 12:01:35 pm: Thank you Ahana ;)