## 4.3 Derivatives of Logarithmic and Exponential Functions

(This topic is also in Section 4.3 in Applied Calculus or Section 11.3 of Finite Mathematics and Applied Calculus)

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The derivatives of the logarithmic functions are given as follows.

Derivative of logb and ln

 ddx logb(x) = 1x ln(b)

An important special case is this:

 ddx ln(x) = 1x Since ln e = 1
Example

 ddx log3(x) = 1x ln(3)

Q Where do these formulas come from?
A For derivations of these formulas, consult Section 4.3 in Applied Calculus, or Section 11.3 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 proper graphing calculator format to input your answers (spaces are ignored). Here are some examples of correctly formatted expressions involving logarithms and exponentials.

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

Q We know how to differentiate with 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. Her is a list of chain rule items from the preceding tutorial with a new item added.

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 logb(x) = 1x ln(b)
 ddx logb(u) = 1u ln(b) dudx
 ddx ln x = 1x
 ddx ln (u) = 1u dudx
The Above Rule in Words:

The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity.

 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

Here is one more for you to try.

Note Use proper graphing calculator format to input your answer (spaces are ignored). Here are some examples of correctly formatted expressions involving logarithms and exponentials.

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

Derivatives of Exponential Functions

The derivatives of the exponential functions are given as follows.

Derivative of bx and ex

 ddx bx = bx ln(b)

An important special case is this:

 ddx ex = ex Since ln(e) = 1
Example

 ddx 2x(4x) = 2(4x) + 2x (4x)ln 4 Product rule

Q Where do these formulas come from?
A Consult Section 4.3 of Applied Calculus, Section 11.3 of Finite Mathematics and Applied Calculus.

These formulas allow us to further expand our table of derivatives:

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 logb(x) = 1x ln(b)
 ddx logb(u) = 1u ln(b) dudx
Logarithm with arbitary base
 ddx ln x = 1x
 ddx ln (u) = 1u dudx
Natural Logarithm
The Above Rule in Words:

The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity.
 ddx bx = bx ln(b)
 ddx bu = bu ln(b) dudx
 ddx ex = ex
 ddx eu = eu dudx
The Above Rule in Words:

The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity.

If you wish to print this 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 4.3 of Applied Calculus, or Section 11.3 of Finite Mathematics and Applied Calculus.

Last Updated: March, 2007