PURE MATHEMATICS - Differential Calculus

 

Exponentials, Logarithms

 

 

exponential ex

logarithms

problems y=Nf(x)

graphs

 

 

 

Exponential Functions

 

Strictly speaking all functions where the variable is in the index are called exponentials.

 

 

The Exponential function e x

 

This is the one particular exponential function where 'e' is approximately 2.71828 and the gradient of y = e x at (0,1) is 1.

 

 

e graphs compared

 

 

One other special quality of y = e x is that its derivative is also equal to e x,

 

 

 

 

and for problems of the type y = e kx :

 

 

 

Derivative problems like the above concerning 'e' are commonly solved using the Chain Rule.

 

 

Example #1

 

Find the derivative of:

 

 

 

Example #2

 

Find the derivative of:

 

 

 

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Derivative of a Natural Logarithm Function

 

Remember y = log e x means:

 

 

x is the number produced when e is raised to the power of y

 

 

The connection between y = e x and y = log e x can be shown by rearranging y = log e x.

 

 

y = log e x   can be written as   x = e y

 

 

(log e x is now more commonly written as ln(x) )

 

 

The derivative of ln(x) is given by:

 

 

 

 

 

e and log graphs compared

 

 

Example #1

 

Find the derivative of y = ln(3x).

 

 

 

Example #2

 

Find the derivative of y = ln(x3+3).

 

 

 

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Problems of the type y = N f(x)

 

Problems of this type are solved by taking logs on both sides and/or using the Chain Rule.

 

 

Example #1

 

Find the derivative of y = 10 x.

 

 

 

Example #2

 

Find the derivative of y = ln(cos32x).

 

 

 

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A Graphical Comparison of Exponential and Log Functions

 

As you can see, y = e x is reflected in the line y = x to produce the curve y = ln(x).

 

 

e reflected in y=x

 

 

 

 

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