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