Derivatives

Question 1

The Difference Quotient

(slope of the secant line)

Answer 1

( ƒ(x₁ + h) – ƒ(x₁) ) / _h

Question 2

The definition of the derivative

(the slope of the tangent line)

Answer 2

ƒ’(x₁) =

lim_h→0 ( ƒ(x₁ + h) – ƒ(x₁) ) / _h

Question 3

d/dx [u•v]

(The Product Rule)

Answer 3

u’ • v + v’ • u

Question 4

d/dx [u/v]

(The Qutient Rule)

Answer 4

(u’ • v - v’ • u) / _v²

Question 5

d/dx ƒ(g(x))

(The Chain Rule)

Answer 5

ƒ’(g(x)) • g’(x)

Question 6

If y = y(v) and v = v(x), then dy/dx =

Answer 6

dy/dx = dy/dv • dv/dx

Question 7

d/dx [ku]

(k = constant, u = variable)

Answer 7

k • du/dx

Question 8

d/dx [k]

Answer 8

0

Question 9

d/dx k/u

Answer 9

-k / u² du/dx

Question 10

d/dx k√u

Answer 10

k / ( 2 √u ) du/dx

Question 11

d/dx [a^u]

Answer 11

a^u • ( ln(a) ) du/dx

Question 12

d/dx e^u

Answer 12

e^u du/dx

Question 13

d/dx [ln u]

Answer 13

1 / u du/dx

Question 14

d/dx [log_bu]

Answer 14

1 / ( u ln(b) ) du/dx

Question 15

d/dx [sin(u)]

Answer 15

cos(u) du/dx

Question 16

d/dx [cos(u)]

Answer 16

-sin(u) du/dx

Question 17

d/dx [tan(u)]

Answer 17

sec²(u) du/dx

Question 18

d/dx [cot(u)]

Answer 18

-csc²(u) du/dx

Question 19

d/dx [sec(u)]

Answer 19

sec(u) • tan(u) du/dx

Question 20

d/dx [cscu]

Answer 20

-cscu•cotu du/dx

Question 21

d/dx [sin^-1(u)]

Answer 21

1 / √ ( 1 – u² ) du/dx

Question 22

d/dx [cos^-1 (u)]

Answer 22

-1 / √ ( 1 – u² ) du/dx

Question 23

d/dx [tan^-1(u)]

Answer 23

1 / ( 1 + u² ) du/dx

Question 24

d/dx [cot^-1(u)]

Answer 24

-1 / ( 1 + u² ) du/dx

Question 25

d/dx [sec^-1(u)]

Answer 25

1 / ( |u| √( u² – 1) ) du/dx

Question 26

d/dx [csc^-1(u)]

Answer 26

-1 / ( |u| √( u² – 1) ) du/dx

Question 27

The derivative of an inverse function d/dx ƒ^-'(x)|_x=c =

Answer 27

1 / [d/dy ƒ(y)] _x=a

Question 28

Parametric equations ^dy/dt / _dx/dt =

Answer 28

dy/dx

Question 29

Formula for differentials

Answer 29

ƒ(x+△x) ≈ ƒ(x) + ƒ'(x)•△x

Question 30

d²y / dx² = ? ## Footnote Second derivative of a parmetric function.

Answer 30

( x' y" – x" y' ) / (x')³

Question 31

Mean value theorem

Answer 31

For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ƒ'(c) = ^{ƒ(b)–ƒ(a)}/_b–a

Question 32

Rolle's theorem

Answer 32

If a real-valued function ƒ is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a *c* in the open interval (a, b) such that ƒ'(c) = 0