Derivatives of Logarithmic and Exponential Functions
(This topic is also in Section 4.5 in Applied Calculus 5e and Section or Section 11.5 of Finite Mathematics and Applied Calculus 5e)
Derivatives of Logarithmic Functions
The derivatives of the logarithmic functions are given as follows:
Derivative of log_{b} and ln
\frac{d}{dx} \log_b(x) = \frac{1}{x\.\ln b}
An important special case is this:
\frac{d}{dx} \ln x = \frac{1}{x\.\ln e} = \frac{1}{x} Because \ln e = 1

Quick Examples
\frac{d}{dx} \log_3(x) = \frac{1}{x\.\ln 3}
\frac{d}{dx}[4\.\ln x] = 4\frac{d}{dx}[\ln x] = 4\frac{1}{x} = \frac{4}{x}
\frac{d}{dx}[4\.\log_3(x)] = 4\frac{d}{dx}[\log_3(x)] = 4\frac{1}{x\.\ln 3} = \frac{4}{x\.\ln 3}
\frac{d}{dx}[x^2\ln x] = 2x\.\ln x + x^2\frac{1}{x} = 2x\.\ln x + x
By the product rule
Some for you
Note Use graphing calculator format to input your answers (spaces are ignored). Here are some examples of expressions involving logarithms and exponentials:
Mathematical Expression 
Input Formula 








log_{3}(x+1)
2x^{2} 

ln(x+1) / ( ln(3) * 2*x^2 ) 


Q Where do these formulas come from?
A For derivations of these formulas, consult Section 4.5 in Applied Calculus, or Section 11.5 in Finite Mathematics and Applied Calculus.
Q
We know how to differentiate expressions that contain 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. Here is a list of chain rule items that include some involving logarithms.
\frac{d}{dx} = ?
\frac{d}{dx} = ?
Derivative of Log of the Absolute Value
Something curious happens if we take the derivative of the logarithm of the absolute value of
x:
\frac{d}{dx} \ln x  = 
\frac{1}{x} \frac{d}{dx} x  
By the chain rule above 
 = 
\frac{1}{x} \frac{x}{x}  
The derivative of x is x/x 
 = 
\frac{1}{x}  
Exactly the same as the derivative of ln x! 
In other words, replacing
x by the absolute value of
x has absolutely no effect on the derivative of the natural logarithm. Similarly it has no effect on the derivative of the logarithm of
x to any base, or on the logarithm of any quantity. Example:
\frac{d}{dx}\ln3x^2x = \frac{1}{3x^2x}(6x1) = \frac{6x1}{3x^2x}
as though the absolute value wasn't there! Thus the answers to all the questions above with "ln () replaced by "ln " are exactly the same. See the textbook
Applied Calculus for more details.
Derivatives of Exponential Functions
The derivatives of the exponential functions are given as follows.
Derivative of b^{x} and e^{x}
\frac{d}{dx} b^x = b^x \ln b
An important special case is this:

Quick Examples
\frac{d}{dx}[3^x] = 3^x \ln 3
\frac{d}{dx}[2e^x] = 2\frac{d}{dx}[e^x] = 2e^x
\frac{d}{dx}[x^2e^x] = 2x\.e^x + x^2e^x = e^x(2x + x^2)
By the product rule
Some for you

Q Where do these formulas come from?
A Consult Section 4.5 of Applied Calculus, Section 11.5 of Finite Mathematics and Applied Calculus.
These formulas allow us to further expand our table of derivatives involving the chain rule:
In this question, several of the choices are based on actual answers students gave in a test. Only one is correct!
\frac{d}{dx} = ?
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Last Updated: October 2009
Copyright © 2009 Stefan Waner