## 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 logb 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
 ln x
 ln(x)
 (4x+2) ln x
 (4x+2)*ln(x)
 ln(ex + 1)
 ln(e^x + 1)
 log3x
 ln(x)/ln(3)
 log3(x+1) 2x2
 ln(x+1) / ( ln(3) * 2*x^2 )

 \frac{d}{dx}[] = \frac{d}{dx}[] = \frac{d}{dx}[] = \frac{d}{dx}\Bigleft(\Bigright) =

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(x2-3x+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)
 d dx f(x) = g(x)
 d dx f(u) = g(u) du dx
General form of the Chain Rule
 d dx xn = nx n-1
 d dx un = nun-1 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
 d dx ln (u) = 1 u du dx
The derivative of the natural logarithm of a quantity is the
reciprocal of that quantity, times the derivative of that quantity.
 d dx logb(x) = 1 x ln(b)
 d dx logb(u) = 1 u ln(b) du dx

Quick Examples

 \frac{d}{dx}\ln(3x^2-x) = \frac{1}{3x^2-x}\frac{d}{dx}[3x^2-x] = \frac{1}{3x^2-x}(6x-1) = \frac{6x-1}{3x^2-x} \frac{d}{dx} = \frac{d}{dx} = \frac{d}{dx} =

\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}\ln|3x^2-x| = \frac{1}{3x^2-x}(6x-1) = \frac{6x-1}{3x^2-x}
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 bx and ex

\frac{d}{dx} b^x = b^x \ln b

An important special case is this:

\frac{d}{dx} e^x = e^x
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

 \frac{d}{dx}[] = \frac{d}{dx}[] =

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:

Original Rule
Generalized Rule
(Chain Rule)
 d dx f(x) = g(x)
 d dx f(u) = g(u) du dx
General form of
Chain Rule
 d dx xn = nx n-1
 d dx un = nun-1 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
 d dx ln (u) = 1 u du dx
The derivative of the natural logarithm of a quantity is the
reciprocal of that quantity, times the derivative of that quantity.
 d dx logb(x) = 1 x ln(b)
 d dx logb(u) = 1 u ln(b) du dx
 d dx ex = ex
 d dx eu = eu du dx
The derivative of e raised to a quantity is e raised to
that quantity, times the derivative of that quantity.
 d dx bx = bx ln(b)
 d dx bu = bu ln(b) du dx

Quick Examples

 \frac{d}{dx}[e^{-x}] = e^{-x}\frac{d}{dx}[-x] = -e^{-x} \frac{d}{dx}[e^{3x2-x}] = e^{3x2-x}\frac{d}{dx}[3x2-x] = (6x-1)e^{3x2-x}

Some for you

 \frac{d}{dx}[] = \frac{d}{dx}[] = \frac{d}{dx}[] =

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