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Derivatives of Trigonometric Functions

General Differentiation

Function

Derivative

sinx

cosx

cosx

-sinx

sin2x

2∙sinxcosx = sin2x

cos2x

-2∙sinxcosx = - sin2x

tanx = sec2x

1/(cos2x) = 1+tan2x

cotx = -csc2x

-1/(sin2x) = -1-cot2x

secx

secxtanx

cscx

-cscxcotx

arcsinx = sin-1x

1/√(1-x2)

arccosx = cos-1x

-1/√(1-x2)

arctanx = tan-1x

1/(1+x2)

arccotx = cot-1x

-1/(1+x2)

arcsecx = sec-1x

1/(|x|∙√(x2-1))

arccscx = csc-1x

-1/(|x|∙√(x2-1))


The following table summarizes the derivatives of the six trigonometric functions, as well as their chain rule counterparts (that is, the sine, cosine, etc. of a function).



Example 1:

Example 2: Find the derivative of y = 3 sin3 (2x4 + 1).

Put u = 2x4 + 1 and v = sin u

So y = 3v3




Example 3: Differentiate

Apply the quotient rule first, then we have


Now apply the product rule in the first part of the numerator, the result of g'(x) will be:




Example 4: Differentiate y = cos3(tan(3x)).

Apply the chain rule four times:




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