Section 4.3: Derivatives of Logarithmic and Exponential Functions

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Derivatives of Logarithmic Functions

The derivatives of the logarithmic functions are given as follows.

Derivative of logb and ln d dx logb(x) = 1 x ln(b) An important special case is this: d dx ln(x) = 1 x

Example d dx log3(x) = 1 x ln(3)

Question
Where do these formulas come from?

Answer
Consult p. 291 of Calculus Applied to the Real World, or p. 789 inFinite Mathematics and Calculus Applied to the Real World.

d dx	(x2+3x)ln x = ?

	2x+3 x			(2x+3)ln(1/x)
	(2x+3)ln x + x + 3			(2x+3)ln x + (x2+3x)ln(1/x)

Question
We know how to differentiate with the logarithm of x. What about the logarithm of a more complicated quantity, for instance ln(x²3x+2)?

Answer
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)
d dx f(x) = g(x)	d dx f(u) = g(u) du dx	General form of Chain Rule
d dx xn = nx n1	d dx un = nun1 du dx	Generalized Power Rule
d dx 4x1 = 4x2	d dx 4u1 = 4u2 du dx	Example
d dx sin x = cos x	d dx sin u = cos u du dx	(Press herefor text on trig functions.)
d dx logb(x) = 1 x ln(b)	d dx logb(u) = 1 u ln(b) du dx
d dx ln x = 1 x	d dx ln (u) = 1 u du dx
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.

d dx	ln(x2+2x1) = ?

	2(x+1) x2+2x1			1 2(x+1)
	ln (2x + 2)			2(x+1) ln(x2+2x1)

d dx

ln(3x+2) 3x+2

= ?

	1 3x+2			3(1 ln(3x+2)) (3x+2)2
	1 x			1 3ln(3x+2) (3x+2)2

Derivatives of Exponential Functions

The derivatives of the exponential functions are given as follows.

Derivative of bx and ex d dx bx = bx ln(b) An important special case is this: d dx ex = ex (since ln(e) = 1)

Example d dx 2x(4x) = 2(4x) + 2x (4x)ln 4 (product rule)

Question
Where do these formulas come from?

Answer
Consult p. 292 of Calculus Applied to the Real World, or p. 790 inFinite Mathematics and Calculus Applied to the Real World.

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

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 n1	d dx un = nun1 du dx	Generalized Power Rule
d dx 4x1 = 4x2	d dx 4u1 = 4u2 du dx	Example
d dx sin x = cos x	d dx sin u = cos u du dx	(Press herefor text on trig functions.)
d dx logb(x) = 1 x ln(b)	d dx logb(u) = 1 u ln(b) du dx	(Logarithm with arbitary base)
d dx ln x = 1 x	d dx ln (u) = 1 u du dx	(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.
d dx bx = bx ln(b)	d dx bu = bu ln(b) du dx
d dx ex = ex	d dx eu = eu du dx
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.

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d dx

[

e4x22

]

= ?

	8x e4x22			e8x
	(4x22)e4x23			e4x22

In the next quiz question, the wrong choices were actual answers obtained from a class test.

d dx

ex ex ex + ex

?

	(11/x)(ex+ex) (exex)(1+1/x) (ex + ex)2			e + e2x e e2x
	4 (ex + ex)2			ex + ex ex ex

Now try some of the exercises on pp. 293-296 of Calculus Applied to the Real World, or pp. 791-794 of Finite Mathematics and Calculus Applied to the Real World.