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21 June, 09:03

Derivative for tan (x)

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  1. 21 June, 09:11
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    sec² (x)

    Explanation:

    Use the trigonometric ratio definition of the tangent function and the quotient rule.

    tan (x) = sin (x) / cos (x)

    Quotient rule: the derivative of a quotient is:

    [the denominator * the derivative of the numerator less the numerator * the derivative of the denominator] / [denominator]²

    (f/g) ' = [ g*f' - f*g'] / g²

    So,

    tan (x) ' = [ sin (x) / cos (x) ]' [ sin (x) / cos (x) ]' = [ cos (x) sin (x) ' - sin (x) cos (x) ' ] / [cos (x) ]²

    = [ cos (x) cos (x) + sin (x) sin (x) ] / [ cos (x) ]²

    = [ cos² (x) + sin² (x) ] / cos² (x)

    = 1 / cos² (x)

    = sec² (x)

    The result is that the derivative of tan (x) is sec² (x)
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