unit 3

composite functions: for f(g(x)), find f(x) and g(x)
1. sin(x^2)
2. ?lnx
3. cos(sin(5x))

1. f(x) = sinx; g(x) = x^2
2. f(x) = ?x; g(x) = lnx
3. for this one it is f(g(h(x))): f(x) = cosx; g(x) = sinx; h(x) = 5x

the chain rule

aka: derivative of a composite function
(d/dx)f(g(x)) = f'(g(x))*g'(x)

find the derivative
1. f(x) = (x^2 - 5)^4
2. g(x) = ?(4x - 3)
3. h(x) = sin^2(5x)
4. y = ln(x^3)
5. y = ln(x^3)
6. f(x) = ((t^2 + 1) / (2t - 5))^3

1. f'(x) = 8x(x^2 - 5)^3
2. g'(x) = 2/?4x - 3
3. h'(x) = 10sin5xcos5x
4. dy/dx = 3/x
5. dy/dx = 3/x
6. f'(x) = (6(t^2 + 1)^2 * (t^2 - 5t - 1)) / (2t - 5)^4

7. if g(x) = 2x?1 - x. find g'(-3)

g'(-3) = 11/2

8. given the following table values. find f'(4) for each function
x: 3, 4
g(x): -1, 3
g'(x): 7, -2
h(x): -2, 9
h'(x): -3, 5
f(x) = (g(x))^2
f(x) = ?h(x)
f(x) = h(g(x))

f'(4) = -12
f'(4) = 5/6
f'(4) = 6

find the derivative of each function
1. g(x) = (3x^2 - 1)^5
2. y = sin2x
3. h(r) = ^3?(5r^2 - 2r + 1)
4. y = ?(4 - cos(x^2))
5. h(x) = ln(5^x)
6. g(x) = ln(2x^3)
7. f(x) = ?(tan(2x))
8. y = cos^2(x)
9. y = 1/((7x^2 - 1)^2)
10. f(x) = 3^(?x)
11. y = sin^3(

1. g'(x) = 30x(3x^2 - 1)^4
2. dy/dx = 2cos2x
3. h'(r) = (10r - 2)/(3*^3?((5r^2 - 2r + 1)^2))
4. dy/dx = (2xsinx^2)/(2?4-cosx^2)
5. h'(x) = ln5
6. g'(x) = 3/x
7. f'(x) = (2sec^2(2x)) / (2?tan2x)
8. dy/dx = -2cosxsinx
9. dy/dx = -28x / (7x^2 - 1)^3
10. f'(x

find f'(5) given the following:
x: 5, 9
g(x): 9, 2
g'(x): 6, -3
h(x): 5, -4
h'(x): -4, 1
15. f(x) = h(g(x))
16. f(x) = (h(x))^2
17. f(x) = ?g(x)
18. f(x) = 2g(x)h(x)
19. f(x) = 1 / h(x)
20. f(x) = g(h(x))

15. 6
16. -40
17. 1
18. -12
19. 4/25
20. -24

find the slope of the tangent line at the given x-value
21. h(x) = (3x - 4)^2 / x at x = -2
22. g(x) = cos(tanx) at x = ?
find the equation of the tangent line at the given x-value
24. f(x) = ?(x^2 - 9) at x = 5
25. g(x) = e^(x^2) at x = 1
26. y = sin^2(3

21. 5
22. 0
24. y - 4 = 5/4(x - 5)
25. y - e = 2e(x - 1)
26. y - (1/2) = -3(x - ?/4)

27. the graph of the function f is shown at the right (the graph has a slope of 1/2 and a y intercept at 3.5)
the function h is defined by h(x) = f(2x^2 - x). find the slope of the tangent line to the graph of h at the point where x = -1

y - 2 = 5/2(x + 1)

28. let f(x) = 2e^3x and g(x) = 5x^3. at what value of x do the graphs of f and g have parallel tangents?
29. let f be the function given by f(x) = 5e^3x^3. for what positive value of a is the slope of the line tangent to the graph of f at (a, f(a)) equal

28. -0.366
29. 0.344
30. 1/16
31. -16

explicit equation example

y = x + 16

implicit equation example

x^2 + y^2 = 16

chain rule and implicit differentiation:
in terms of x:
(d/dx)x =
(d/dx)x^2 =
(d/dx)e^5x =
in terms of y:
(d/dx)y =
(d/dx)y^2 =
(d/dx)e^5y =

1
2x
e^(5x)*5
dy/dx
2y(dy/dx)
e^(5y)*5(dy/dx)

implicit differentiation example: find dy/dx for y^2 - 5x^3 = 3y

dy/dx = 15x^2 / 2y - 3

1. y^3 - 2x = x^4 + 2y
2. sin(xy) = 10x

dy/dx = (4x^3 + 2) / (3y^2 - 2)
dy/dx = (10sec(xy) - y) / x

find the equation of all tangent lines for x^2 + y^2 = 4 when x = 1

y - ?3 = (-1/?3)(x-1)
y + ?3 = (1/?3)(x-1)

horizontal tangent lines exist when the slope, dy/dx =
vertical tangent lines exist when the slope, dy/dx =

0
undefined
vertical tangent lines can also be undefined because of a cusp or corner

4. find all horizontal tangent lines of the graph 3x^2 + 2y^2 = 16
5. find all vertical tangent lines of the graph 3x^2 + 2y^2 = 16

4. y = +/-?8
5. x = +/-?(16/3)

find dy/dx
1. 5x^2 + 2y^3 = 4
2. 5y^2 + 3 = x^2
3. sin(x + y) = 2x
4. 4x + 1 = cosy^2
5. 5x^2 - e^(4y^2) = -6
6. ln(y^3) = 5x + 3
7. x^2 = 4y^3 + 5y^2
8. 5x^3 - 2y = 5y^3
9. lny^2 + cos^2(x) = 1 - y
10. sin(y/2) + e^y = 4x
11. x^3 + y^3 = 6xy
12. x / siny

1. dy/dx = -5x / 3y^2
2. dy/dx = x / 5y
3. dy/dx = 2sec(x + y) - 1
4. dy/dx = -2csc(y^2) / y
5. dy/dx = 5x / 4ye^(4y^2)
6. dy/dx = 5y/3
7. dy/dx = x / 6y^2 + 5y
8. dy/dx = 15x^2 / 15y^2 + 2
9. dy/dx = 2cosxsinx / ((2/y) + 1)
10. dy/dx = 4 / ((1/2)cos(1/2)

find the slope of the tangent line at the given point
14. 2 = 3x^4 + xy^4 at (-1,1)
15. xlny = 4 - 2x at (2,1)
find the equation of the tangent line at the given point
16. x^2 + y^2 + 19 = 2x + 12y at (4,3)
17. xsin2y = ycos2x at (?/4, ?/2)
find the equat

14. -11/4
15. -1
16. y - 3 = x - 4
17. y - ?/2 = 2(x - ?/4)
18. horizontal: y = +/-2.031; vertical: x = -3.372 and 2.372
19. horizontal: none; vertical: x = -1/4

20. find the slope of the normal line to y = x + cos(xy) at (0,1)
21. the graph of f(x), shown below, consists of a semicircle and two line segments. the semi circle crosses the x axis at -2 and 2, and the y axis at 2, the line segments create a point at

20. -1
21. -1/?3
22. +/-2/3

a function's inverse is found by...

swapping the input (x) and output (y) values

reciprocal notation:
x^-1 =

1/x

inverse notation:
f^(-1)(x) means...

inverse of f

three ways to say the same thing about inverses:

1. g(x) is the inverse of f(x)
2. g(x) = f^(-1)(x)
3. f(g(x)) = x and g(f(x)) = x

derivative of an inverse function:

(d/dx)[f^(-1)(x)] = 1 / (f'[f^(-1)(x)])

the table below gives the values of the differentiable functions f, g, and f' at selected values of x. let g(x) = f^(-1)(x)
x: 1, 2, 3, 4
f(x): 3, 1, -5, 0
f'(x): -3, -2, -5, -6
1. what is the value of g'(1)
2. write an equation for the line tangent to f^

1. -1/2
2. y - 2 = -1/2(x - 1)
3. -1/2
4. 1/36

for each problem, let f and g be differentiable functions where g(x) = f^(-1)(x) for all x.
1. f(3) = -2, f(-2) = 4, f'(3) = 5, and f'(-2) = 1. find g'(-2)
2. f(1) = 5, f(2) = 4, f'(1) = -2, and f'(2) = -4. find g'(5)
3. f(6) = -2, f(-3) = 7, f'(6) = -1,

1. 1/5
2. -1/2
3. 1/3
4. 1/7

the table below gives the values of the differentiable function g and its derivative g' at selected values of x. let h(x) = g^(-1)(x).
x: -1, -2, -3, -4, -5
g(x): -2, -5, -4, -3, -1
g'(x): -4, -2, -1, -5, -3
5. find h'(-1); find the equation of the tangen

5. -1/3; y + 5 = -1/3(x + 1)
6. -1/5; y + 4 = -1/5(x + 3)
7. -1/2; y + 2 = -1/2(x + 5)

f and g are differentiable functions. use the table to answer the problem below. f and g are not inverses!
x: 1, 2, 3, 4, 5, 6
f(x): 5, 1, 6, 2, 3, 4
f'(x): -5, -6, 4, 9, 1, 2
g(x): 4, 3, 1, 6, 1, 2
g'(x): 5, 3, 6, 1, 2, 4
8. g^(-1)(4)
9. f^(-1)(5)
10. d/

8. 1
9. 1
10. 1/3
11. -1/6
12. y - 4 = 1/9(x - 2)

for each function g(x), its inverse g^(-1)(x) = f(x). evaluate the given derivative
13. g(x) = cos(x) + 3x^2; g(?/2) = 3?/4; find f'(3?/4)
14. g(x) = 2x^3 - x^2 - 5x; g(-2) = -10; find f'(-10)
15. g(x) = ?(8 - 2x). find f'(4)
16. g(x) = x^3 - 7. find f'(2

13. 1/(3? - 1)
14. 1/23
15. -4
16. 1/27
17. -20

the functions f and g are differentiable for all real numbers and g is strictly increasing. the table below gives values of the functions and their first derivatives at selected values of x. the function h is given by h(x) = f(g(x)) - 6.
x: 1, 2, 3, 4
f(x

a. h(1) = 3
h(3) = -7
because of the IVT, h(r) = -5
b. y - 1 = 1/5(x-2)

a function h satisfies h(3) = 5 and h'(3) = 7. which of the following statements about the inverse of h must be true?

(h^-1)'(5) = 1/7

1. h(x) = ((x+5) / (x^2 + 2))^2; h'(x) =
2. g(x) = sin(tan(2x)); g'(x) =
3. x^2 - y^2 = 25; dy/dx =
4. x^3 + y^3 = 6xy - 1; dy/dx =
5. equation of line tangent to (y - 3)^2 = 4(x - 5) at (6,1)

1. h'(x) = (2((x + 5)/(x^2 + 2))^2)*(((x^2 + 2) - (2x^2 + 10x)) / (x^2 + 2)^2)
2. g'(x) = 2cos(tan(2x))(sec^2(2x))
3. dy/dx = x/y
4. dy/dx = (6y - 3x^2) / (3y^2 - 6x)
5. y - 1 = -x + 6

1. inverse of f(x) = 2x. name that function g(x)
a. g(x) =
b. f(3) =
c. g(6) =
d. f'(x) =
e. g'(x) =
f. relationship between g'(x) and f'(x)
2. find inverse of f(x) = x^2. name that function g(x)
a. f(2) =
b. g(4) =
c. f'(2) =
d. g'(4) =
e. relationship b

1. a. x/2
b. 6
c. 3
d. 2
e. 1/2
f. reciprocals
2. g'(x) = ?x
a. 4
b. 2
c. 4
d. 1/4
e. reciprocals
3. x
4. g'(x) = 1 / f'(g(x)) --- derivative of inverse of a function definition

notation: the inverse of a trig function x may be indicated using the...

inverse function f^(-1) or with the prefix "arc" (ex: sin^(-1)x = arcsinx)

inverse trig derivatives:
(d/dx)sin^(-1)(x) =
(d/dx)sec^(-1)(x) =
(d/dx)tan^(-1)(x) =
(d/dx)cos^(-1)(x) =
(d/dx)csc^(-1)(x) =
(d/dx)cot^(-1)(x) =

(d/dx)sin^(-1)(x) = 1/?(1 - x^2)
(d/dx)sec^(-1)(x) = 1/(IxI?(x^2 - 1))
(d/dx)tan^(-1)(x) = 1/(x^2 + 1)
(d/dx)cos^(-1)(x) = -1/?(1 - x^2)
(d/dx)csc^(-1)(x) = -1/(IxI?(x^2 - 1))
(d/dx)cot^(-1)(x) = -1/(x^2 + 1)

if it starts with "s"...

it has subtraction and a square root
in addition: if the "i" is the second letter, the 1 is first; if the "e" is the second letter, the 1 is second
if the "c" is third, aww crap there is an absolute value

find the derivative
1. (d/dx)sin^(-1)(3x)
2. (d/dx)tan^(-1)(2x^2)
3. (d/dx)arcsec(5x)

1. 3/?(1 - 9x^2)
2. 4x/(4x^4 + 1)
3. 5/(IxI?(25x^2 - 1)

simplify the following expressions.
4. 9x^2/(I3x^3I?(9x^6 - 1))
5. 4x/(I2x^2I?(4x^2 - 1))

4. 3/(IxI?(9x^6 - 1))
5. 2/(x?(4x^2 - 1))

domain and range of an inverse trig function
y = sin^(-1)(x)
y = cos^(-1)(x)
y = tan^(-1)(x)

*look at pictures/graph on first page of 3.4
domain: -1<=x<=1; range: -?/2<=y<=?/2
domain: -1<=x<=1; range: 0<=y<=?
domain: -infinity<=x<=infinity; range: -?/2<y<?/2 (open circle = <>)

evaluate each function at the given x-value.
6. f(x) = arcsinx at x = ?3/2
7. f(x) = cos^(-1)(x/4) at x = -2
8. f(x) = arctan at x = 1/?3

6. ?/3
7. 2?/3
8. ?/6

find the derivative of each expression
1. d/dxsin^(-1)(5x)
2. d/dxcsc^(-1)(4x^5)
3. d/dxarctan(2x)
4. d/dxsec^(-1)(x^3)
5. d/dxcsc6x
6. d/dxarcos(3x^2)
7. d/dxcot^(-1)(-x)
8. d/dxcos^(-1)(-7x)
9. d/dxarccsc(x^6)
10. d/dxcot^(-1)(4x^4)

1. 5/?(1 - 25x^2)
2. -5/(IxI?(16x^10 - 1))
3. 2/(4x^2 + 1)
4. 3/(IxI?(x^6 - 1))
5. -1/(IxI?(36x^2 - 1))
6. -6x/(?(1 - 9x^4))
7. 1/(x^1 + 1)
8. 7/(?(1 - 49x^2))
9. -6/(x?(x^12 - 1))
10. -16x^3/(16x^8 + 1)

find the tangent line equation of the curve at the given point
11. y = arcsin(x) at the point where x = ?2/2
12. y = cos^(-1)(4x) at the point where x = ?3/8
13. y = arctan(3x^2) at the point where x = ?3/3
14. y = sin^(-1)(5x) at the point where x = -?3/

11. y - ?/4 = ?2(x - ?2/2)
12. y - ?/6 = -8(x - ?3/8)
13. y - ?/4 = ?3(x - ?3/3)
14. y + ?/3 = 10(x + ?3/10)
15. y - 3?/4 = -?2(x + ?2/2)
16. y - ?/3 = 1/4(x - ?3)

let g(x) = (arccosx^2)^5. then g'(x) =

-10((x(arccosx^2)^4)/(?(1-x^4)))

if lim h--^ 0 arccos(a+h) - arccos(a) / h = 3, which of the following could be the value of a?

?8 / 3

if arctany = lnx, then dy/dx =

1+y^2 / x

what procedures for finding the derivative have you learned so far this year?
Unit 2:
Unit 3:

Unit 2: powder rule, constant, constant multiple, sum/difference, trig, exponential, logarithm, product rule, quotient rule
Unit 3: chain rule, implicit differentiation, inverse, inverse trig

1. if f(x) = x^(2)lnx, then f'(x) =
2. if f and g are functions such that f(g(x)) = x for all x in their domains, and if f(a) = b and f'(a) = c, then which of the following is true?
3. find the equation of the tangent line to 9x^2 + 16y^2 = 52 through (2,

x + 2xlnx
g'(b) = 1/c
9x - 8y - 26 = 0
1
-6/(3x-1)^2
1/(4?(x)?(1+?x))
2
5^(2x)/x + 2ln(5)ln(3x)5^(2x)
1/10
1/(?1-x^2) + 1/?x

notation: y, f(x), y
1st derivative:
2nd derivative:
3rd derivative:
nth derivative:

y', f'(x), dy/dx
y'', f''(x), d^(2)y/dx^2
y''', f'''(x), d^(3)y/dx^3
y^(n), f^(n)(x), d^(n)y/dx^n

finding the 2nd derivative
f(x) = x^6 - 2x^4 + 5x^2 - 3x + 9
f'(x) =
f''(x) =
f'''(x) =
f^(4)(x) =

6x^5 - 8x^3 + 10x - 3
30x^4 - 24x^2 + 10
120x^3 - 48x
360x^2 - 48

finding the 2nd derivative:
y = ?x + x^(-2)
dy/dx =
d^(2)y/dx^2 =

1/2?x - 2/x^3
-1/4x^(-3/2) + 6x^-4

2nd derivative with implicit differentiation
find d^(2)y/dx^2 for siny = x + y

siny/(cosy - 1)^3

Find d^2y/dx^2
1. y = sinx + ln(5x)
2. y = e^xlnx
3. y = sin^2x
4. dy/dx = y^2 + 2x - 1
5. dy/dx = 1/y - 3x
6. dy/dx = xy^2
7. sin(x+y) = 2x
8. e^x = y^3 + 1
9. lny = 5x + 3
10. if f(x) = -3x^3 + 4x^(-2), find f''(-2)
11. if f(x) = xlnx, find f''(1)
12. i

-sinx - 1/x^2
e^xlnx + 2e^x/x - e^x/x^2
2cos^2x - 2sin^2x
2y^3 + 4xy - 2y + 2
-1/y^3 + 3x/y^2 - 3
y^2 + 2x^2y^3
4sec^2(x+y)tan(x+y)
(e^x3y^2 + (2e^2x)/y) / 9y^4
25y
75/2
1
-35/32
-25
-51
1 + e^2

find the derivative of the following:
f(x) = 3x^7 - 4x^3 + 5x
f'(x) =
f''(x) =
f'''(x) =
f^(4)(x) =

21x^6 - 12x^2 + 5
126x^5 - 24x
630x^4 - 24
2520x^3

find the derivative of the following:
y = 4?x
dy/dx =
d^2y/dx^2 =

2x^(-1/2)
-1/?x^3

find the derivative of the following:
y = 1/x^3 - 1/2x^4
y' =
y'' =
y''' =

-3x^-4 - 2x^3
12x^-5 - 6x^2
-60x^-6 - 12x

given f(x) = 3x^2 - x + 2, g(x) = 1/x^3, and h(x) = ?x. find the following
20. f''(2)
21. g'''(-3)
22. 2h''(4)

6
-20/243
-1/16

if f(x) = (1 + x/20)^5, find f''(40)

1.350

a curve given by the equation x^3 = xy = 8 has slope given by dy/dx = -3x^2 - y / x. the value of d^2y/dx^2 at the point where x = 2 is

0

if y = xe^x, then d^ny/dx^n =

(x + n)e^x

let g be the function given by g(x) = cos(-x) - sinx + 6. which of the following statements is true for y = g(x)?

y - 6 = d^4y/dx^4

find the derivative
1. h(x) = cos^2(4x)
2. y = ln?x+3
3. x^2 + 2y^5 = 10xy
4. y = csc^(-1)(x^3)

-8cos4xsin4x
1/2x+6
x-5y/5x-5y^4
-3/(IxI?(x^6-1))

for each problem, let f and g be differentiable functions where g(x) = f^-1(x) for all x.
5. f(6) = -1, f(4) = -2, f'(6) = 3, f'(4) = 7. whats the value of g'(-1)
6. let f be the function defined by f(x) = x^3 + 3x + 1. let g(x) = f^-1(x), where g(-3) = -

1/3
1/6

find d^2y/dx^2 based on the given information
7. y = x^5 - e^4x
8. y = y^2 + x
9. find the equation of the tangent line. x^2 + 7y^2 = 8y^3 at (-6,2)
10. if x = y^2 - cosx find d^2y/dx^2 at (?/6,1/2)

20x^3 - 16e^4x
2/(1-2y)^3
y - 2 = -3/17(x+6)
(-?3 - 1) / 2