Derivative Rules

Question 1

Instantaneous Rate of Change of f(x)

Answer 1

lim h–>0 f(x+h)-f(x)/h
or
lim x–>a f(x) - f(a)/x - a

Question 2

-CONSTANT RULE-
Find f’(x)
f(x)= k {when k is a constant}

Answer 2

f’(x)= 0

Question 3

-CONSTANT MULTIPLE RULE-
Find g’(x)
g(x)= kx {when k is a constant}

Answer 3

g’(x)= k

Question 4

-POWER RULE-
Find r’(x)
r(x)= xⁿ

Answer 4

r’(x)= nxⁿ⁻¹

Question 5

Differentiability implies _________

Answer 5

continuity.

Question 6

Functions are not differentiable at _________

Answer 6

sharp turns, cusps, corners, and vertical tangents.

Question 7

Functions are differentiable when:

Answer 7

1) it is continuous at that point
2) lim x–>a-…f’(x) = lim x–>a+…f’(x)

Question 8

-CONSTANT MULTIPLE RULE-
Find h’(x)
h(x)= k[f(x)] {when k is a constant}

Answer 8

h’(x)= k[f’(x)]

Question 9

-SUM/DIFFERENCE RULE-
Find j’(x)
j(x)= f(x) +- g(x)

Answer 9

j’(x)= f’(x) +- g’(x)

Question 10

Find z’(x)
z(x)= sin(x)

Answer 10

z’(x)= cos(x)

Question 11

Find q’(x)
q(x)= cos(x)

Answer 11

q’(x)= -sin(x)

Question 12

Find p’(x)
p(x)= eˣ {when e is Euler’s number}

Answer 12

p’(x)= eˣ

Question 13

Find m’(x)
m(x)= ln(x)

Answer 13

m’(x)= 1/x

Question 14

-QUOTIENT RULE-
Find t’(x)
t(x)= f(x) / g(x)

Answer 14

t’(x)= [(g(x))(f’(x))] - [(f(x))(g’(x))] / [g(x)]²

Question 15

-PRODUCT RULE-
Find h’(x)
h(x)= f(x)*g(x)

Answer 15

h’(x)= f(x)g’(x) + g(x)f’(x)

Question 16

Find h’(x)
h(x)= tan(x)

Answer 16

h’(x)= [sec(x)]²

Question 17

Find k’(x)
k(x)= cot(x)

Answer 17

k’(x)= [-csc(x)]²

Question 18

Find m’(x)
m(x)= sec(x)

Answer 18

m’(x)= sec(x)*tan(x)

Question 19

Find p’(x)
p(x)= csc(x)

Answer 19

p’(x)= cot(x)*-csc(x)

Question 20

Evaluate lim x–>a f(x) - f(a)/x-a using derivatives.

Answer 20

f’(a)