Calculus Maximus WS 4.10: Log Functions Page 1 of 8 Name_________________________________________ Date________________________ Period______ Worksheet 4.10—Derivatives of Log Functions & LOG DIFF Show all work. No calculator unless otherwise stated. 1. Find the derivative of each function, given that a is a constant (a) a y x = (b) x y a = (c) x y x = (d) a y a = 2. Find the derivative of each. Remember to simplify early and often (a) 2ln x d e dx = (b) sin log x a d a dx = (c) 5 2 log 8 x d dx − = 3. For each of the following, find dy dx . Remember to “simplify early and often.” (a) 3 1 log 2 x x y − = (b) 3/2 2 log 1 y x x = +
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Calculus Maximus WS 4.10: Log Functions
Page 1 of 8
Name_________________________________________ Date________________________ Period______ Worksheet 4.10—Derivatives of Log Functions & LOG DIFF Show all work. No calculator unless otherwise stated. 1. Find the derivative of each function, given that a is a constant (a) ay x= (b) xy a= (c) xy x= (d) ay a= 2. Find the derivative of each. Remember to simplify early and often
(a) 2ln xd edx⎡ ⎤ =⎣ ⎦ (b) sinlog x
ad adx⎡ ⎤⎣ ⎦ = (c) 5
2log 8xddx
−⎡ ⎤⎣ ⎦=
3. For each of the following, find dydx
. Remember to “simplify early and often.”
(a) 31log
2x xy −
= (b) 3/ 22log 1y x x= +
Calculus Maximus WS 4.10: Log Functions
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(c) coslncos 1
xyx
=−
(d) 1ln lnyx
⎛ ⎞= ⎜ ⎟⎝ ⎠
(e) 3lny x= (f) 2lny x x=
(g) ( )3log 1 lny x x= + (h) 4 4 2ln3 1xyx−
=+
Calculus Maximus WS 4.10: Log Functions
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4. Use implicit differentiation to find dydx
.
(a) 2 23ln 10x y y− + = (b) ln 5 30xy x+ = 5. Find an equation of the tangent line to the graph of ( )21 ln 2x y x y+ − = + at ( )1,0 .
6. A line with slope m passes through the origin and is tangent to ln3xy ⎛ ⎞= ⎜ ⎟
⎝ ⎠. What is the value of m?
Calculus Maximus WS 4.10: Log Functions
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7. Find an equation for a line that is tangent to the graph of xy e= and goes through the origin. 8. Use LOG DIFF:
(a) ( ) ( )
( )
4 2
53
3 1
2 5
x xddx x
⎡ ⎤− +⎢ ⎥⎢ ⎥
−⎢ ⎥⎣ ⎦
(b) If 1/ ln xy x= , find dydx
.
Calculus Maximus WS 4.10: Log Functions
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9. Let ( ) ( )2ln 1f x x= − .
(a) State the domain of f. (b) Find ( )f xʹ′ . (c) State the domain of ( )f xʹ′ . (d) Prove that ( ) 0f xʹ′ʹ′ < for all x in the domain of f.
Calculus Maximus WS 4.10: Log Functions
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Multiple Choice 10. Use the properties of logs to simplify, as much as possible, the expression: