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The derivatives of the logarithmic functions are given as follows.
Derivative of log_{b} and ln
An important special case is this:


Example

Q Where do these formulas come from?
A For derivations of these formulas, consult Section 4.3 in Applied Calculus, or Section 11.3 in Finite Mathematics and Applied Calculus.
Here are more for you to try.
Note Use proper graphing calculator format to input your answers (spaces are ignored). Here are some examples of correctly formatted expressions involving logarithms and exponentials.
Q
We know how to differentiate with the logarithm of x. What about the logarithm of a more complicated quantity, for instance ln(x^{2}3x+2)?
A
To differentiate something like that, we need to use the chain rule. Her is a list of chain rule items from the preceding tutorial with a new item added.
(Chain Rule) 
Comments  


General form of Chain Rule 



Generalized Power Rule  


An example of the above rule  


Take me to text on trig functions!  






The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity. 
Q  dx 
ln(x^{2}+2x1) = ? 
Q  dx 
3x+2 
= ? 
Here is one more for you to try.
Note Use proper graphing calculator format to input your answer (spaces are ignored). Here are some examples of correctly formatted expressions involving logarithms and exponentials.
Derivatives of Exponential Functions
The derivatives of the exponential functions are given as follows.
Derivative of b^{x} and e^{x}
An important special case is this:


Example

Q Where do these formulas come from?
A Consult Section 4.3 of Applied Calculus, Section 11.3 of Finite Mathematics and Applied Calculus.
These formulas allow us to further expand our table of derivatives:
(Chain Rule) 
Comments  


General form of Chain Rule 



Generalized Power Rule  


An example of the above rule  


Take me to text on trig functions!  


Logarithm with arbitary base  


Natural Logarithm  
The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity. 







The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity. 
Q  dx 
[  e^{4x22}  ]  = ? 
In the next quiz question, all the choices were actual answers students gave in a test. Only one is correct!
Q  dx 
e^{x} + e^{x}  ? 






Now try some of the exercises in Section 4.3 of Applied Calculus, or Section 11.3 of Finite Mathematics and Applied Calculus.