General Differentiation
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Function
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Derivative
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sinx
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cosx
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cosx
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-sinx
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sin2x
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2∙sinx∙cosx = sin2x
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cos2x
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-2∙sinx∙cosx = - sin2x
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tanx = sec2x
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1/(cos2x) = 1+tan2x
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cotx = -csc2x
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-1/(sin2x) = -1-cot2x
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secx
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secx∙tanx
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cscx
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-cscx∙cotx
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arcsinx = sin-1x
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1/√(1-x2)
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arccosx = cos-1x
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-1/√(1-x2)
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arctanx = tan-1x
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1/(1+x2)
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arccotx = cot-1x
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-1/(1+x2)
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arcsecx = sec-1x
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1/(|x|∙√(x2-1))
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arccscx = csc-1x
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-1/(|x|∙√(x2-1))
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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:
