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
Find the derivative of each function
1) f(x) = e3x f '(x) = 3e3x
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
Use rule two above!
g(x) = x2-4x |
g'(x) = 2x - 4 |
4) f(x) = ln 5x Use rule 4 above!
f '(x) = 5/5x = 1/x
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)
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?)