,
Calculus: dy/dx 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 ex

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

e graphs compared

One other special quality of y= ex is that its derivative is also equal to ex

derivation of e to the power x

and for problems of the type y= ekx

derivative of e kx

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

Example #1

Find the derivative of:

answer to example #1

Example #2

find the derivative of:

e f(x0 problem #2

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Derivative of a Natural Logarithm function

Remember y=logex means:

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

The connection between y=ex and y=logex can be shown by rearranging y=logex.

y=logex   can be written as   x=ey

(logex is now more commonly written as ln(x) )

The derivative of ln(x) is given by:

derivative of log x

e and log graphs compared

Example #1

find the derivative of y = ln(3x)

e problem #1

Example #2

find the derivative of y = ln(x3+3)

log e problem #2

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Problems of the type y=Nf(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=10x

N to the power f(x) problem #1

Example #2

find the derivative of y= ln(cos32x)

log problem #2

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A graphical comparison of exponential and log functions

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

e reflected in y=x

 

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