Trigonometry Differentiation Rules

A derivative of a function is the rate of change of the function or the slope of the line at a given point. The derivative of f(a) is notated as f ( a ) or d dx f( a ) .

This discussion will focus on the basic trigonometric differentiation rules.

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS:

FUNCTION

DERIVATIVE

FUNCTION

DERIVATIVE

d dx sinx

cos x

d dx cscx

-csc x cot x

d dx cosx

-sin x

d dx secx

sec x tan x

d dx tanx

sec2 x

d dx cotx

-csc2 x




Let's look at some examples:

To work these examples requires the use of various differentiation rules. If you are not familiar with a rule go to the associated topic for a review.


20tan x

Step 1: Apply the Constant Multiple Rule.


d dx [ cf( x ) ]=c d dx f( x )

20 d dx tanx Constant Mul.

Step 2: Take the derivative of tan x.

20sec2 x Tangent Rule

Example 1:      sinx cosx

Step 1: Apply the product rule.


d dx [ f( x )g( x ) ]=f( x ) d dx [ g( x ) ]+g( x ) d dx [ f( x ) ]

d dx [ sin x cos x ]


sin x d dx cos x+cosx d dx sin x

Step 2: Take the derivative of each part.


Apply the appropriate trigonometric differentiation rule.

d dx cos x


sinx Sine Rule


__________________________


d dx sin x

cosx   Cosine Rule

Step 3: Substitute the derivatives & simplify.

sin x(sinx)+( cos x )(cosx) 


sin 2 x+ cos 2 x

Example 2:       1+sin x cos x

Step 1: Apply the quotient rule.


d dx [ f( x ) g( x ) ]= g( x ) d dx [ f( x ) ]f( x ) d dx [ g( x ) ] [ g( x ) ] 2

d dx [ 1+sinx cosx ]


cosx  d dx (1+sinx)(1+sinx) d dx cosx cos 2 x

Step 2: Take the derivative of each part.


Apply the appropriate trigonometric differentiation rule.

d dx 1+sinx Original


d dx 1+ d dx sinx       Sum Rule


0 + cos x Constant/Sine


cos x

__________________________


d dx cosx   Original


-sin x       Sine Rule


-sin x

Step 3: Substitute the derivatives & simplify.

cosx (cosx)(1+sinx)  (sinx) cos 2 x


cos 2 x+sinx+ sin 2 x cos 2 x


1+sinx cos 2 x





Related Links:
Math
algebra
Inverse Trigonometric Differentiation Rules
Logarithmic Differentiation
Calculus Topics


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