## 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 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 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 ddx (x2+3x)ln x = ?
 2x+3x
(2x+3)ln(1/x)
(2x+3)ln x + x + 3 (2x+3)ln x + (x2+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
 ln x
 ln(x)
 x ln x
 x*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 )

Q
 ddx x ln(x)
 =
Q
 ddx x log5x
 =

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)
 ddx f(x) = g(x)
 ddx f(u) = g(u) dudx
General form of
Chain Rule
 ddx xn = nx n-1
 ddx un = nun-1 dudx
Generalized Power Rule
 ddx 4x-1/2 = -2x-3/2
 ddx 4u-1/2 = -2u-3/2 dudx
An example of the above rule
 ddx sin x = cos x
 ddx sin u = cos u dudx
Take me to text on trig functions!
 ddx ln x = 1x
 ddx ln (u) = 1u dudx
The derivative of the natural logarithm of a quantity is the
reciprocal of that quantity, times the derivative of that quantity.
 ddx logb(x) = 1x ln(b)
 ddx logb(u) = 1u ln(b) dudx

Here is one for you to try.

Q
 ddx ln(3x2 - 1/x)
 =

 Q ddx ln(x2+2x-1) = ?

2(x+1)

x2+2x-1
 12(x+1)
ln (2x + 2)
2(x+1)

ln(x2+2x-1)

 Q ddx ln(3x+2)3x+2 = ?

 13x+2 3(1 - ln(3x+2))(3x+2)2 1x 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}\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 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:

Original Rule
Generalized Rule
(Chain Rule)
 ddx f(x) = g(x)
 ddx f(u) = g(u) dudx
General form of
Chain Rule
 ddx xn = nx n-1
 ddx un = nun-1 dudx
Generalized Power Rule
 ddx 4x-1/2 = -2x-3/2
 ddx 4u-1/2 = -2u-3/2 dudx
An example of the above rule
 ddx sin x = cos x
 ddx sin u = cos u dudx
Take me to text on trig functions!
 ddx ln x = 1x
 ddx ln (u) = 1u dudx
The derivative of the natural logarithm of a quantity is the
reciprocal of that quantity, times the derivative of that quantity.
 ddx logb(x) = 1x ln(b)
 ddx logb(u) = 1u ln(b) dudx
 ddx ex = ex
 ddx eu = eu dudx
The derivative of e raised to a quantity is e raised to
that quantity, times the derivative of that quantity.
 ddx bx = bx ln(b)
 ddx bu = bu ln(b) dudx

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}

If you wish to print the above table out, press here to get a new page showing the table by itself.

 Q ddx [ e4x2-2 ] = ?

 8x e4x2-2 e8x (4x2-2)e4x2-3 e4x2-2

In the next quiz question, all the choices were actual answers students gave in a test. Only one is correct!

 Q ddx ex - e-xex + e-x ?

 (1-1/x)(ex+e-x) - (ex-e-x)(1+1/x)(ex + e-x)2
 e + e-1e - e-1
 4(ex + e-x)2
 ex + e-xex - 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