The first derivation and maximum summarize

Question

f(x)=1/x2∗2e^−2x

 

 

Answer 1

1) If f(x) = ex, then f '(x) = ex 
2) If f(x) = eg(x), then f '(x) = g'(x).eg(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

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Find the derivative of each function

1)  f(x) = e3x                                 f '(x) = 3e3x

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2)  f(x) = x3ex         Use the product rule!

r(x) = x3	s(x) = ex
r'(x) = 3x2	s'(x) = ex

f '(x) = x3ex + 3x2ex = (x3 + 3x2)ex

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           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 = 10 + ln x10   ( log rules, remember?)

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