4.3 Derivatives of Logarithmic and Exponential Functions

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

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 4.3 in Applied Calculus, or Section 11.3 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)	Comments
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 4.3 of Applied Calculus, Section 11.3 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)	Comments
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: June, 2008 Copyright © 2008 Stefan Waner