Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Language: English

Derivative Rules Compilation
The theorems below are considered the first step towards understanding calculus
...
This
document will only list the theorems without their proofs since the proofs would make the document
lengthy and would be better if placed in another document
...

The numbering of the theorems is based on the order they appear during the lesson
...
Take note, however, that the theorems labeled as numbers 34 and 35 are miscellaneous and are
placed at the end because they would not fit with the usual numbering
...
Unless otherwise specified, it is 𝑥
𝑑

as seen in 𝑑𝒙
...


Theorem 1: Constant Rule
For any constant 𝑐:
𝑑
𝑐=0
𝑑𝑥

Theorem 2: Power Rule of Derivatives
For 𝑛 ∈ ℝ:
𝑑 𝑛
𝑥 = 𝑛𝑥 𝑛−1
𝑑𝑥

Theorem 3: Constant Multiple
Given a constant 𝑐 and a function 𝑓 in terms of 𝑥:
𝑑
𝑐𝑓(𝑥) = 𝑐𝑓′(𝑥)
𝑑𝑥

Theorem 4: Derivative of the Sum and Difference of Two Functions
Given two functions 𝑓and 𝑔 in terms of 𝑥:
𝑑
[𝑓(𝑥) ± 𝑔(𝑥)] = 𝑓 ′ (𝑥) ± 𝑔′(𝑥)
𝑑𝑥
Provided 𝑓′(𝑥) and 𝑔′(𝑥) exist
...


Theorem 6: Quotient Rule of Derivatives
Given two functions 𝑓and 𝑔 in terms of 𝑥, and 𝑔(𝑥) ≠ 0
...


Theorem 7: Derivative of the Sine Function
𝑑
(sin 𝑥) = cos 𝑥
𝑑𝑥

Theorem 8: Derivative of the Cosine Function
𝑑
(cos 𝑥) = − sin 𝑥
𝑑𝑥

Theorem 9: Derivative of the Tangent Function
𝑑
(tan 𝑥) = sec 2 𝑥
𝑑𝑥

Theorem 10: Derivative of the Cotangent Function
𝑑
(cot 𝑥) = − csc 2 𝑥
𝑑𝑥

Theorem 11: Derivative of the Secant Function
𝑑
(sec 𝑥) = tan 𝑥 sec 𝑥
𝑑𝑥

Theorem 12: Derivative of the Cosecant Function
𝑑
(csc 𝑥) = − csc 𝑥 cot 𝑥
𝑑𝑥

Theorem 13: Fermat’s Theorem
If 𝑓 has a relative extremum at a number 𝑐, then 𝑐 must be a critical value of 𝑓
...
If 𝑓(𝑎) = 𝑓(𝑏) = 0, then 𝑓 ′ (𝑐) = 0 for at least one
number 𝑐 ∈ (𝑎, 𝑏)

Theorem 16: Mean Value Theorem
Suppose 𝑓 is continuous on the closed interval [𝑎, 𝑏] and differentiable on the open
interval (𝑎, 𝑏), then
𝑓 ′ (𝑐) =

𝑓(𝑏) − 𝑓(𝑎)
𝑏−𝑎

*for at least one point in 𝑐 ∈ (𝑎, 𝑏)

Theorem 17: Extended Mean Value Theorem
Suppose 𝑓 and 𝑔 are continuous on the closed interval [𝑎, 𝑏] and differentiable on
the open interval (𝑎, 𝑏)
...
if 𝑓 ′ (𝑥) > 0 ∀𝑥 ∈ (𝑎, 𝑐) and then 𝑓 ′ (𝑥) < 0 ∀𝑥 ∈ (𝑐, 𝑏), then 𝑓(𝑐) is a
relative min
...
if 𝑓 ′ (𝑥) < 0 ∀𝑥 ∈ (𝑎, 𝑐) and then 𝑓 ′ (𝑥) > 0 ∀𝑥 ∈ (𝑐, 𝑏), then 𝑓(𝑐) is a
relative max
...
if 𝑓′(𝑥) has the same sign on the intervals (𝑎, 𝑐) and (𝑐, 𝑏), then 𝑓(𝑐) is
not an extremum
...
718281828459045)

Theorem 22: Derivative of Second Tetration
𝑑 𝑥
𝑥 = 𝑥 𝑥 (1 + ln 𝑥)
𝑑𝑥

Theorem 23: Derivative of Logarithmic Function
For 𝑏 > 0, 𝑏 ≠ 1:
𝑑
1
(log 𝑏 𝑥) =
𝑑𝑥
𝑥 ln 𝑏

Theorem 24: Derivative of the Natural Logarithm
𝑑
1
ln 𝑥 =
𝑑𝑥
𝑥

Theorem 25: Derivative of Inverse Sine
𝑑
1
arcsin 𝑥 =
𝑑𝑥
√1 − 𝑥 2
𝜋 𝜋

Provided arcsin 𝑥 ∈ (− , )
2 2

Theorem 26: Derivative of Inverse Cosine
𝑑
−1
arccos 𝑥 =
𝑑𝑥
√1 − 𝑥 2
Provided arccos 𝑥 ∈ (0, 𝜋)

Theorem 27: Derivative of Inverse Tangent
𝑑
1
arctan 𝑥 =
𝑑𝑥
1 + 𝑥2
𝜋 𝜋

Provided arctan 𝑥 ∈ (− , )
2 2

Theorem 28: Derivative of Inverse Cotangent
𝑑
−1
arccot 𝑥 =
𝑑𝑥
1 + 𝑥2
Provided arccot 𝑥 ∈ (0, 𝜋)

Theorem 29: Derivative of Inverse Secant
𝑑
1
±1
arcsec 𝑥 =
=
𝑑𝑥
|𝑥|√𝑥 2 − 1 𝑥√𝑥 2 − 1
𝜋

Take the positive sign if arcsec 𝑥 ∈ (0, ); take the negative sign if
2
𝜋

arcsec 𝑥 ∈ ( , 𝜋)
2

Theorem 30: Derivative of Inverse Cosecant
𝑑
−1
±1
arccsc 𝑥 =
=
𝑑𝑥
|𝑥|√𝑥 2 − 1 𝑥√𝑥 2 − 1
𝜋

Take the negative sign if arccsc 𝑥 ∈ (0, ); take the positive sign if
2
𝜋

arccsc 𝑥 ∈ (− , 0)
2

Theorem 31: Derivative of Inverse Function
Suppose that 𝑦 = 𝑓(𝑥) and 𝑥 = 𝑓 −1 (𝑦)
...


If 𝑓 ′′ (𝑥) > 0 ∀𝑥 ∈ (𝑎, 𝑏) then the graph of 𝑓 is concave upward in (𝑎, 𝑏)

ii
...
Then 𝑓’’(𝑐) = 0

Theorem 34: 𝑛th derivative of a Power Function
𝑑𝑛 𝑛
𝑥 = 𝑛!
𝑑𝑥 𝑛

Theorem 35: L’Hôpital’s Rule
Given an indeterminate quotient limit lim

𝑓(𝑥)

𝑥→𝑎 𝑔(𝑥)

L’Hôpital’s Rule states that:
𝑓(𝑥)
𝑓′(𝑥)
= lim
𝑥→𝑎 𝑔(𝑥)
𝑥→𝑎 𝑔′(𝑥)
lim

If the resulting quotient limit is still indeterminate, L’Hôpital’s Rule can be applied
over and over until it is no longer indeterminate

---