unit 3

derivative of an inverse function:

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

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)

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

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

(x + n)e^x

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: -π/2<=y<=π/2 domain: -1<=x<=1; range: 0<=y<=π domain: -infinity<=x<=infinity; range: -π/2<y<π/2 (open circle = <>)

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

-10((x(arccosx^2)^4)/(√(1-x^4)))

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

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))

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. if f(x) = 3√x - 32/x, find f''(4) 13. if dy/dx = 3cosy + 5x, find d^2y/dx^2 at (2, π/2) 14. if dy/dx = (4-x)/(2y-3), find d^2y/dx^2 at (-1,2) 15. if dy/dx = lnxe^y, find d^2y/dx^2 at (e,1)

-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

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

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

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)

if arctany = lnx, then dy/dx =

1+y^2 / 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) at x = 1 3. let g be a differentiable function such that g(12) = 4, g(3) = 6, g'(12) = -5, and g'(3) = -2. the function h is differentiable and h(x) = g^(-1)(x) for all x. what is the value of h'(6)? 4. if f(x) = 3x^3 + 1 and g is the inverse function of f, what is the value of g'(25)?

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, and f'(-3) = 3. find g'(7) 4. f(-1) = 4, f(2) = -3, f'(-1) = -5, and f'(2) = 7. find g'(-3)

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

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)

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)

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 between f'(x) and g'(x) 3. if the g(x) is the inverse of f(x), what is f(g(x))? 4. use chain rule to solve for g'(x)

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

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 = 5 13. lnx * e^(3y) = 2y^2

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) + e^y) 11. dy/dx = (2y - x^2) / (y^2 - 2x) 12. dy/dx = tany / x 13. dy/dx = (-e^3y) / (x(3lnxe^3y - 4y))

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

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

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(4x) 12. y = e^(√(1 - cox)) 13. g(x) = e^(cos(7x^3)) 14. h(x) = sin(ln(x^5))

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) = (3^(√x)ln3) / (2√x) 11. 12sin^2(4x)cos(4x) 12. (sinxe^(√(1 - cosx))) / (2√(1 - cosx)) 13. g'(x) = -21x^2sin(7x^3)e^(cos(7x^3)) 14. h'(x) = (5cos(lnx^5)) / x

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

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

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

1.350

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

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. what s the value of g'(-3)?

1/3 1/6

reciprocal notation: x^-1 =

1/x

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 = -√3/10 15. y = arches(x) at the point where x = -√2/2 16. y = arctan(x) at the point where x = √3

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

for each function g(x), its inverse g^(-1)(x) = f(x). evaluate the given derivative 13. g(x) = cos(x) + 3x^2; g(π/2) = 3π/4; find f'(3π/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'(20) 17. g(x) = 5 / (x + 3). find f'(1/2)

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

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 (π/4, π/2) find the equations of all horizontal and vertical tangent lines. calculator allowed (round to 3 decimals) 18. x^2 + x + 2y^2 = 8 19. x + y = y^2

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

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

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 (4,2) (first has a slope of 1 starting at (0,2) and ending that the point, the other line starts at the point and ends at (0,6) with a slope of -1; f'(1) = 22. find the value(s) of dy/dx of x^2y + y^2 = 5 at y = 1

20. -1 21. -1/√3 22. +/-2/3

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 (π/6,1/2)

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

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 = π 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(3x) at x = π/4

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

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

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 to 6? 30. let f(x) = √2x. if the rate of change of f at x = c is four times its rate of change at x = 1, then c = 31. let f(x) = x*g(h(x)) where g(4) = 2, g'(4) = 3, h(3) = 4, and h'(3) = -2. find f'(3).

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

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

2x^(-1/2) -1/√x^3

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))

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)

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 tangent line to g^(-1) at x = -1 6. h'(-3); find the equation of the tangent line to g^(-1) at x = -3 7. h'(-5); find the equation of the tangent line to g^(-1) at x = -5

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)

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

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. π/3 7. 2π/3 8. π/6

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

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/dxg^(-1)(3) 11. d/dxf^(-1)(1) 12. find the line tangent to the graph of f^(-1)(x) at x = 2

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

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

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): 6, 9, 10, -1 f'(x): 4, 2, -4, 3 g(x): 2, 3, 4, 6 g'(x): 5, 1, 2, 7 a. explain why there must be a value r for 1<r<3 such that h(r) = 5 b. If g^(-1) is the inverse function of g, write an equation for the line tangent to the graph of y = g^(-1)(x) at x = 2

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

the chain rule

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

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

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

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

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

7. if g(x) = 2x√1 - x. find g'(-3)

g'(-3) = 11/2

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

inverse of f

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

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

siny/(cosy - 1)^3

a function's inverse is found by...

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

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,-1) 4. what is the slope of the line tangent to the curve y = arctan(2x) at the point when x = 1/2? 5. if f(x) = (3x^2 + x)/(3x^2 - x), then f'(x) is 6. if f(x) = √(1 + √(x)), find f'(x). 7. a curve is generated by the equation x^2 + 4y^2 = 16. determine the number of points on the curve whose corresponding tangent lines are horizontal 8. d/dx(ln(3x)5^(2x)) = 9. let a function f be defined as f(x) = x^3 - 2x - 4 for x>=1. let g(x) be the inverse function of f(x) and note that f(2) = 0. the value of g'(0) = 10. d/dx(sin^(-1)x + 2√x) =

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

implicit equation example

x^2 + y^2 = 16

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)

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 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)

explicit equation example

y = x + 16

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

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

√8 / 3