Logarithmic Functions

The number e is a transcendental number - beyond any ordinary number, and it is an irrational number. Its interesting uses are utilised by mathematics in a number of analysis, mainly: financial interest, growth and decay of population, and this number is also a part of the topic logarithm as well. The number e happens naturally and many geniuses are baffled about the existence of this number in real life. Its name was given by Leonhard Euler (1707-1783) a prodigy in many studies of nature.

To understand the number e, let's come back to the concept of compound interest:

A=P{ (1+r) }^{ n }

Let's say there is a bank that offers 100% p.a. interest per year. After one year, $1 becomes:

A=1{ (1+1) }^{ 1 }= 2

But what if the bank compounds the interest twice a year? Then the $1 becomes:

A=1{ (1+\cfrac { 1 }{ 2 } ) }^{ 2 }= 2.25

And if it is compounded further...:

A=1{ (1+\cfrac { 1 }{ 3 } ) }^{ 3 }= 2.37\\ \\ A=1{ (1+\cfrac { 1 }{ 4 } ) }^{ 4 }= 2.44\\ \\ A=1{ (1+\cfrac { 1 }{ 5 } ) }^{ 5 }= 2.49

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And if we exponentially use a bigger term:

A=1{ (1+\cfrac { 1 }{ 1000000 } ) }^{ 1000000 }=2.71828...

 

Yes, you probably notice that there is a limit that this growth function operates into. Thus, the number e is heavily linked to growth (exponentially).

 

e = 2.71828..........

 

Derivatives Exponential Functions

 

The function y={e}^{x} is a basic function determining exponential growth. And what is interesting is, e can be applied to calculus: Derivatives and Primitive Functions.

The rate of growth of exponential function is very similar to those of its gradient function. This leads mathematicians to deduce an important nature of exponential function:

\cfrac { d }{ dx } ({ e} ^ { x} )={ e} ^ { x}

 

Examples

 

1. Differentiate 6{e}^{x}.

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\cfrac { d }{ dx } (6{ e}^{ x} )\\ \\ =\quad 6\cfrac { d }{ dx } ({ e}^ { x} )\\ \\ =\quad 6{ e}^ { x}

 

2. Find the equation of the tangent of the curve y=2{e}^{x} at the point (0, 2).

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\cfrac { dy }{ dx } =2{ e }^{ x },\qquad at\quad (0,2):\\ \\ \cfrac { dy }{ dx } =2{ e }^{ 0 }=2\\ \\ Using\quad point-gradient\quad form:\\ y-{ y }_{ 1 }=m(x-{ x }_{ 1 })\\ y-3=3(x-0)\\ y=3x+3

 

In addition, there are also derivatives that uses the function of function rule, product rule and quotient rule. These all apply in any sort of differentiation.

 

3. Differentiate {e}^{-4x+3}.

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y'=f'(x){e}^{f(x)} \\ y'=-4{e}^{-4x+3}

 

4. Differntiate {x}^{3}{e}^{2x}.

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Using product rule:

\cfrac { dy }{ dx } =u'v+v'u\\ \\ \cfrac { dy }{ dx } =3x\cdot { e }^{ 2x }+2{ e }^{ 2x }\cdot { x }^{ 2 }\\ \\ \cfrac { dy }{ dx } =x{ e }^{ 2x }(3+2x)

 

5. Differentiate \cfrac{2x+3}{{e}^{x}}.

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\cfrac { dy }{ dx } =\cfrac { u'v-v'u }{ { v }^{ 2 } } \\ \\ \cfrac { dy }{ dx } =\cfrac { 2\cdot { e }^{ x }-{ e }^{ x }(2x+3) }{ { { (e }^{ x } })^{ 2 } } \\ \\ \cfrac { dy }{ dx } =\cfrac { 2{ e }^{ x }-2x{ e }^{ x }-3{ e }^{ x } }{ { e }^{ 2x } } \\ \\ \cfrac { dy }{ dx } =\cfrac { -{ e }^{ x }-2x{ e }^{ x } }{ { e }^{ 2x } } \\ \\ \cfrac { dy }{ dx } =\cfrac { -{ e }^{ x }(1+2x) }{ { e }^{ 2x } } \\ \\ \cfrac { dy }{ dx } =\cfrac { -(1+2x) }{ { e }^{ x } }

 

Integration of Exponential Functions

 

Since the derivative of exponential function is equal to the function, then the reverse is similar. The primitive of an exponential function (indefinite integral) of the exponential function can be derived as:

\int { { e }^{ x }dx= } { e }^{ x }+C

...and...

\int { { e }^{ ax+b }dx= } \cfrac { 1 }{ a } { e }^{ ax+b }+C

 

Examples

 

1. Find \int { { e }^{ 3x+2 }-{ e }^{ x }dx }.

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\int { { e }^{ 3x+2 }-{ e }^{ x }dx } \\ \\ =\quad \cfrac { 1 }{ 3 } { e }^{ 3x+2 }-{ e }^{ x }+C

 

2. Find the exact area enclosed between the curve y={e}^{3x}, the x-axis and the lines x=0 and x=2.

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