# Function examples find y 1 2 1 log y 6 2 x 1 2 ln 6 2

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Chapter P / Exercise 105
Algebra and Trigonometry: Real Mathematics, Real People
Larson
Expert Verified
FUNCTION Examples: Find y’. 1. 2. ) 1 ( log y 6 2 x . ) 1 ( 2 ln 6 2 ln ) 1 ( ) 1 ( y' 6 5 6 6 x x x x d x x 2 ln y 3 ) ( 2 ln ) 2 (ln y' 3 3 x xd x d x x x x x x x x 2 ln 3 ) 3 ( 2 ln 2 2 y' 2 2 2 3 ) 2 ln 3 1 ( y' 2 x x
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Chapter P / Exercise 105
Algebra and Trigonometry: Real Mathematics, Real People
Larson
Expert Verified
A. Find the derivative of each of the following natural logarithmic functions and simplify. 1 x 4 ln y . 1 3 2 1 log ) ( . 2 x x f x 5 3 1 x 3 ln x f . 3 2 x 3 x ln x h . 4 4 x x x G 3 log ) 1 ( . 5 3 2 x ln 1 x 4 ln x g . 6 x ln x ln y . 7 2 2 2 x 1 x 1 x ln x x F . 8
B. Differentiate the following logarithmic functions. x 3 log y . 1 2 2 2 2 x log 3 log x h . 2 2 2 y 3 log y f . 3 2 x x 3 log x g . 4 3 2 3 4 x log x F . 5 1 z z 3 4 log z H . 6 2 5  1 t 3 3 t 2 3 t log t G . 7 2 2
C. Find the derivative using implicit differentiation. y x 5 xy ln . 1 1 ln ln . 2 y x y x 2 4y - logx xlny 3.
The EXPONENTIAL FUNCTION . . y log x as written be also may a y function, c logarithmi of inverse the is function l exponentia the Since number. real a is x where a y by defined is 1, a and 0 a a, base with function l exponentia The a x x
. Laws of Exponents n m a n a m a . 1 n m if , m - n a 1 n m if , 1 n m if , n m a n a m a . 2 mn a n m a . 3 n b n a n ab . 4 n b n a n b a . 5
Law of Exponents 0 a provided , 1 0 a . 6 n 1 m a m n 1 a n m a . 7 x a . 8 x log a y x then a a if . 9 y x
DIFFERENTIATION FORMULA Derivative of Exponential Function The derivative of the exponential function for any given base and any differentiable function of u. f(x) u where ; dx du e ) e ( dx d f(x) u where ; dx du a ln a ) a ( dx d u u u u : e base For : a base given any For
DERIVATIVE OF EXPONENTIAL FUNCTIONS Examples: Find y’. 1. 2. 8 6 3 y 1 4 x 2 x e x 6 ) 4 ( ) 2 ( 3 ln 3 y' 1 4 x 2 x e x ]. 3 2 3 ) ( 3 [ln 2 y' 1 4 2 x x e x 8 ) 2 - (1 y x 2 ln ) 2 ( ) 2 - (1 8 y' 7 x x . ) 2 - )(1 2 ( 2 ln 8 y' 7 x x
A. Find the derivative of each of the following and simplify the result: EXAMPLE : 2 x 3 e x f . 1 x 2 1 e x g . 2 x / 1 2 e x 4 x h . 3 2 y x 2 x xy e . 4 5 x 4 2 x 3 7 y . 5 2 x 3 4 ln x h . 6 7. G x log e x 1 e 2 x 3 8. f x 2 3 x 4 5 x 2 9. 3 x 5 y x 4 y
A. Find the derivative and simplify the result. B. Apply the appropriate formulas to obtain the derivative of the given function and simplify. EXERCISES : 1 x 3 x 2 3 x g . 1 2 2 x ln x e x f . 2 2 e e y . 3 x 3 x 4 3 x 2 2 2 3 x log x h . 4 2 x 5 x G . 1 2 ln y x xe ye . 2 2 2 y x 2 X 1 x x H . 3 1 x 2 e y . 4 1 x 2 x 2 x 2 e e ln x f . 5