Derivatives of the Exponential and Log Functions
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    Exponential and Log Derivatives 

1) If f(x) = e^x, then f '(x) = e^x
2) If f(x) = e^g(x), then f '(x) = g'(x)^.e^g(x)
3) If f(x) = ln x, then f '(x) = 1/x   (x > 0)

4) If f(x) = ln g(x), then f '(x) = g'(x)/g(x)    [g(x) > 0]

Sample Problems

1)  f(x) = e^3x                                 f '(x) = 3e^3x

2)  f(x) = x³e^x         Use the product rule!

          Use rule two above!

g(x) = x2-4x

g'(x) = 2x - 4




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4)  f(x) = ln 5x                       Use rule 4 above!

g(x) = 5x

g'(x) = 5

f '(x) = 5/5x = 1/x


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5)  f(x) = ln ( 3x² - 5x)          Again, use rule 4 above!

g(x) = 3x2 - 5x

g'(x) = 6x - 5

f '(x) = (6x - 5)/(3x² - 5x)


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6)  f(x) = 5x ln x²            Use the product rule and rule 4 above!

r(x) = 5x	s(x) = ln x2
r'(x) = 5	s'(s) = 2x/x2 = 2/x

f '(x) = 5x(2/x) + 5 ln x²
                                       = 10 + ln x¹⁰   ( log rules, remember?)

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7)  f(x) = (e^5x)(ln x)         Use the product rule and rules 2 and 3!!

r(x) = ex	s(x) = ln x
r'(x) = ex	s'(x) = 1/x

f '(x) = e^x/x + e^x(ln x)
= e^x(1/x + ln x)

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                Use the quotient rule and rule 1 above!


 

r(x) = ex	s(x) = 2x + 3
r'(x) = ex	s'(x) = 2



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That's about it for derivatives!!  Next section is integration!!

Current quizaroo #  C1

 Find each derivative by using the appropriate rule!

1)  f(x) = 3x²(5x² + 6)  Find f'(x)

a) 15x⁴ + 18x²

b) 60x³ + 36x

c) 60x²

d) 18x

e) 15x³

 

2)  f(x) = (3x + 7)/(5x + 8)  Find f'(x)
 

a) 15

b) 0

c) 3/5

d) -11/(5x + 8)²

e) 3/(5x + 8)²

3)  f(x) = (3x - 1)(4x + 2)/(2x)  Find f'(x)

a) 6

b) 6/x

c) (6x² + 1)/x²

d) 12x²/(x + 1)

e) derivative does not exist!

 

4)  f(x) = (2x + 5)⁶  Find f'(x)

a) 12(2x + 5)⁵

b) 12

c) 64

d) 6(2x)⁵

e) 0

5)  f(x) = (2x² - 3)e^x

a) 2e^x

b) 3e^x

c) 4xe^x - 3e^x

d) (2x² - 3)e^x + 4xe^x

e) ln x