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# Derivatives of Trigonometric Functions

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General Differentiation

 Function Derivative sinx cosx cosx -sinx sin2x 2∙sinx∙cosx = sin2x cos2x -2∙sinx∙cosx = - sin2x tanx = sec2x 1/(cos2x) = 1+tan2x cotx = -csc2x -1/(sin2x) = -1-cot2x secx secx∙tanx cscx -cscx∙cotx 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: