Derivatives of Functions

Question 1

f(x) = ln(x)
f'(x) = ?

https://youtu.be/3LgfZ4bQ-yc

Answer 1

1
─
x

Question 2

f(x) = e^x 
f'(x) = ?

https://youtu.be/SFWN-TkVFyI

Answer 2

e^x

Question 3

f(x) = sin(x) 
f'(x) = ?

Answer 3

cos(x)

Question 4

f(x) = cos(x) 
f'(x) = ?

Answer 4

-sin(x)

Question 5

Product Rule
g(x) = f(x) h(x)
g’(x) = ?

https://youtu.be/L5ErlC0COxI

Answer 5

f’(x) * h(x) + f(x) * h’(x)

Question 6

Quotient Rule
f(x) =

g(x)
───
h(x)

f’(x) = ?

Answer 6

g’(x)h(x) - g(x)h’(x)
──────────
(h(x))^2

Question 7

f(x) =

sin(x)
───
x

limit of f(x) = ?

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/modal/v/sinx-over-x-as-x-approaches-0

Answer 7

1

Question 8

d/dx tan(x) = ?

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/modal/v/derivatives-of-tanx-and-cotx

Answer 8

1
────
cos^2(x)

or
sec^2(x)

Question 9

d/dx cot(x) = ?

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/modal/v/derivatives-of-tanx-and-cotx

Answer 9

-1
─────
sin^2(x)

or
-csc^2(x)

Question 10

d/dx sec(x) = ?

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/modal/v/derivatives-of-secx-and-cscx

Answer 10

sin(x) / cos^2(x)
or 
tan(x) / cos(x)
or
tan(x) sec(x)

Question 11

d/dx csc(x) = ?

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/modal/v/derivatives-of-secx-and-cscx

Answer 11

-cos(x) / sin^2(x)
or
-(cot(x) csc(x))

Question 12

Sine Sum Identity
sin(a + b) = ?

https://www.youtube.com/watch?v=zcEMKv5yIYs

Answer 12

sin(a) cos(b) + sin(b) cos(a)

Question 13

Sine Difference Identity
sin(a - b) = ?

https://www.youtube.com/watch?v=zcEMKv5yIYs

Answer 13

sin(a) cos(b) - sin(b) cos(a)

Question 14

Cosine Sum Identity
cos(a + b) = ?

https://www.youtube.com/watch?v=zcEMKv5yIYs

Answer 14

cos(a) cos(b) - sin(a) sin(b)

Question 15

Cosine Difference Identity
cos(a - b) = ?

https://www.youtube.com/watch?v=zcEMKv5yIYs

Answer 15

cos(a) cos(b) + sin(a) sin(b)

Question 16

Tan Sum Identity
tan(a + b) = ?

https://www.youtube.com/watch?v=zcEMKv5yIYs

Answer 16

tan(a) + tan(b)
─────────
1 - tan(a) tan(b)

Question 17

Tan Difference Identity
tan(a - b) = ?

https://www.youtube.com/watch?v=zcEMKv5yIYs

Answer 17

tan(a) - tan(b)
─────────
1 + tan(a) tan(b)

Question 18

Chain Rule
g(x) = f( h(x) )
g’(x) = ?

Answer 18

f’( h(x) ) * h’(x)

Question 19

d/dx

a^x = ?

Answer 19

ln a (a^x)

Question 20

d/dx

log_a (x) = ?

Answer 20

1
────
ln(a) x

Question 21

d/dx of inverse functions

Let h(x) be the inverse of f(x)

h’(x) = ?

Answer 21

1
————
f’(h(x))

Question 22

d/dx arcsin(x)

Answer 22

1
——————-
Sqrt (1 - x^2)

Hint: Derivation involves taking the sine of both sides and substituting using the trig sum

Question 23

d/dx = arccos(x)

Answer 23

-1
—————-
Sqrt( 1-x^2 )

Question 24

d/dx of arctan(x)

Answer 24

1
———-
1 + x^2

Question 25

Critical Point

Answer 25

A point where the slope is 0 or undefined. | This indicates when the slope has changed directions which hints at relative min/max points

Question 26

Mean Value Theorem

Answer 26

f(y) - f(x) ———— y-x ^^^^^^^ let this be m If the graph is continuous and differentiable then there is a point that has the slope m in the domain [x, y]

Question 27

L'Hôpital's rule

Answer 27

lim as x --> c for f(x) f'(x) —— = —— g(x) g'(x) You can use this formula recursively until you get a value