Chapter 4: Products, Quotients, and Parametric Functions

Contains vocabulary and concepts from Chapter 4 of Calculus: Concepts and Applications by Paul A. Foerster as taught by Colin Suehring at McFarland High School

the product rule

if y = uv, where u and v are differentiable functions of x, then y’= u’v + uv’

derivative of the first times second, plus first times derivative of second

the quotient rule

if y = u/v, where u and v are differentiable functions, and v doesn’t = 0, then y’ = (u’v = uv’)/v²

the quotient rule rhyme

low d high, minus high d low, square the bottom, and away we go!

sin’x

cosx

cos’x

-sinx

tan’x

sec²x

cot’x

-csc²x

sec’x

secx tanx

csc’x

-cscx cotx

tanx

sinx/cosx

cotx

cosx/sinx

secx

1/cosx

cscx

1/sinx

sin²x + cos²x =

1

1 + cot²x =

csc²x

tan²x + 1 =

sec²x

derivative of the inverse function

if y = f⁻¹(x), then d/dx (f-1(x)) =

1/ (f’(f⁻¹(x))

d/dx (sin^-1x)

1/√1-x²

d/dx (cos^-1x)

-1/√1-x²

d/dx (tan^-1x)

1/1+x²

d/dx (cot^-1x)

-1/1+x²

d/dx (sec^-1x)

1/|x|√x²-1

d/dx (csc^-1x)

-1/|x|√x²-1

if a function f is differentiable at a point x=c, then

f is continuous at x=c

if a function f is not continuous

it is not differentiable

related rates

D.R.E.D.S.