Flashcard #4: Derivatives - Part B (No Chain Rule)

lim x→0 cos(π/2 + h) - cos(π/2) / h

This equals the derivative of cos(x) evaluated at x = π/2. Therefore the value is -sin(π/2) = -1

lim x→0 3(x + h)² - 3x² / h

This equals the derivative of 3x², which is 6x.

lim △x→0 tan(x + △x) - tan(x) / △x

This equals the derivative of tan(x), which is sec²x.

lim h→0 (4 + h)² - 16 / h

This equals the derivative of x² evaluated at x = 4. Therefore the value is 8.

y = 5

y’ = ?

0

y = x⁷

y’ = ?

7x⁶

y = secx

y’ = ?

secxtanx

y - sinx

y’ = ?

cosx

y = cosx

y’ = ?

-sinx

y = cotx

y’ = ?

-csc²x

y = tanx

y’ = ?

sec²x

y = cscx

y’ = ?

-cotxcscx

y = √x

y’ = ?

1/2(x)^-1/2

y = x²

y’ = ?

2x

d/dx [a^x]

a^xlna

d/dx [e^x]

e^x lne = e^x

y = ln[x]

y’ = ?

1/x

y = log_a x

y’ = ?

y’ = 1/lna • 1/x = 1/xlna

Power Rule:

d/dx (xⁿ) =

nx^x-1

Product Rule:

y = uv

y’ = ?

uv’ + vu”

Quotient Rule:

y = u/v

y’ = ?

vu’ - uv’ / v²

s’(t) =

Velocity or v(t)

v’(t) =

Acceleration or a(t)

Speed =

Velocity

d/dx tan^-1x

1 / x² + 1