Derivatives of Logarithmic and Exponential Functions
(This topic is also in Section 4.5 in Applied Calculus 5e 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

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

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 
d
dx  (x^{2}+3x)ln x = ? 

2x+3
x 


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

 (2x+3)ln x + (x^{2}+3x)ln(1/x) 

Here are more for you to try.
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.
Original Rule 
Generalized Rule (Chain Rule) 
Comments 
d
dx 
f(x) = g(x) 

d
dx 
f(u) = g(u) 
du
dx 

General form of Chain Rule 
d
dx 
x^{n} = nx^{ n1} 

d
dx 
u^{n} = nu^{n1} 
du
dx 

Generalized Power Rule 
d
dx 
4x^{1/2} = 2x^{3/2} 

d
dx 
4u^{1/2} = 2u^{3/2} 
du
dx 

An example of the above rule 
d
dx 
sin x = cos x 

d
dx 
sin u = cos u 
du
dx 

Take me to text on trig functions! 
d
dx 
ln x 
= 
1
x 


The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity. 
d
dx 
log_{b}(x) 
= 
1
x ln(b) 

d
dx 
log_{b}(u)
 = 
1
u ln(b) 
du
dx 


Here is one for you to try.
Q 
d
dx 
ln(x^{2}+2x1) = ? 
 2(x+1)
x^{2}+2x1 

 1
2(x+1) 

 ln (2x + 2) 

 2(x+1)
ln(x^{2}+2x1) 

Q 
d
dx 
ln(3x+2)
3x+2 
= ? 
 1
3x+2 


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

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

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:

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

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:
If you wish to print the above table out, press here to get a new page showing the table by itself.
Q 
d
dx 
[ 
e^{4x22} 
] 
= ? 
 8x e^{4x22} 

 e^{8x} 
 (4x^{2}2)e^{4x23} 

 e^{4x22} 

In the next quiz question, all the choices were actual answers students gave in a test. Only one is correct!
Q 
d
dx 
 e^{x}  e^{x}
e^{x} + e^{x}  
? 

(11/x)(e^{x}+e^{x})  (e^{x}e^{x})(1+1/x)
(e^{x }+ e^{x})^{2} 



e + e^{1}
e  e^{1} 


4
(e^{x }+ e^{x})^{2} 



e^{x} + e^{x}
e^{x } e^{x} 


Now try some of the exercises in Section 4.5 of Applied Calculus, or Section 11.5 of
Finite Mathematics and Applied Calculus.
Last Updated: October 2009
Copyright © 1999, 2003, 2006, 2007, 2009 Stefan Waner