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Basic Calculus Derivative of Trig Funcs Cheat Sheet (DRAFT) by

Basic Calculus Reporting

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Trigon­ometric Identities

Reciprocal Trigon­ometric Identities
Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
Tan θ = 1/Cot θ or Cot θ = 1/Tan θ
Pythag­orean Trigon­ometric Identities
sin2 a + cos2 a = 1
1+tan2 a = sec2 a
cosec2 a = 1 + cot2 a
Ratio Trigon­ometric Identities
Tan θ = Sin θ/Cos θ
Cot θ = Cos θ/Sin θ
Sum and Difference of Angles Trigon­ometric Identities
sin(α+­β)=­sin­(α).co­s(β­)+c­os(­α).s­in(β)
sin(α–­β)=­sin­α.c­osβ­–co­sα.sinβ
cos(α+­β)=­cos­α.c­osβ­–si­nα.sinβ
cos(α–­β)=­cos­α.c­osβ­+si­nα.sinβ

Derivation Formula

Product Rule
(d/dx) (fg)= fg’ + gf’
Quotient Rule
(d/dx) (f/g) = gf'-fg'/g2
Chain Rule
y = f(g(x)), then y' = f'(g(x)). g'(x)
 

Deriva­tives of Trigon­ometric Functions

If f( x) = sin x, then f′( x) = cos x
If f( x) = cos x, then f′( x) = −sin x
If f( x) = tan x, then f′( x) = sec 2 x
If f( x) = cot x, then f′( x) = −csc 2 x.
If f( x) = sec x, then f′( x) = sec x tan x
If f( x) = csc x, then f′( x) = −csc x cot x

Example 1

Example 2

Examples

g ( x ) = 3 sec ( x ) − 10 cot ( x )
 

Derivative of sin x

Derivative of cos x

Derivative of sec x

We will determine the derivative of sec x using the chain rule. We will use the following formulas and identities to calculate the deriva­tive:

sec x = 1/cos x
tan x = sin x/cos x
(cos x)' = -sin x
(sec x)' = (1/cos x)' = (-1/co­s2x­).(cos x)'

= (-1/co­s2x­).(-sin x)

= sin x/cos2x

= (sin x/cos x).(1/cos x)

= tan x sec x

Therefore, d(sec x)/dx = tan x sec x
 

Derivative of cot x

We will determine the derivative of cot x using the quotient rule. We will use the following formulas and identities to calculate the deriva­tive:

(sin x)' = cos x
(cos x)' = -sin x
cot x = cos x/ sin x
cos2x + sin2x = 1
cosec x = 1/sin x
(cot x)' = (cos x/sin x)'

= [(cos x)' sin x - (sin x)' cos x]/sin2x

= [-sin x. sin x - cos x. cos x]/sin2x

= (-sin2x - cos2x)­/sin2x

= -1/sin2x

= -cosec2x

Therefore, d(cot x)/dx = -cosec2x

Derivative of cosec x

We will determine the derivative of cosec x using the chain rule. We will use the following formulas and identities to calculate the deriva­tive:

cosec x = 1/sin x
cot x = cos x/sin x
(sin x)' = cos x
(cosec x)' = (1/sin x)' = (-1/si­n2x­).(sin x)'

= (-1/si­n2x­).(cos x)

= -cos x/sin2x

= -(cos x/sin x).(1/sin x)

= -cot x cosec x

Therefore, d(cosec x)/dx = -cot x cosec x