Limits and Derivatives

Limits and Derivatives

Question

Find the derivatives of the following functions :

space space space space space space space space space space space space space space space space space space fraction numerator straight x over denominator 1 plus tan space straight x end fraction

Answer

Let     space space space space space space space space space space space space fraction numerator straight x over denominator 1 plus tan space straight x end fraction
Differentiating both sides w.r.t. x, we get
space space space space space space space space space space space space space space space space space space space space space space space dy over dx equals fraction numerator left parenthesis 1 plus tan space straight x right parenthesis begin display style straight d over dx end style straight x minus straight x begin display style straight d over dx end style left parenthesis 1 plus tan space straight x right parenthesis over denominator blank end fraction equals space fraction numerator left parenthesis 1 plus tan space straight x right parenthesis space 1 minus straight x left parenthesis 0 plus sec squared straight x right parenthesis over denominator left parenthesis 1 plus tan space straight x right parenthesis squared end fraction
                  space space space space space space space space space space space space space space space space space equals space space fraction numerator 1 plus tan space straight x space minus space straight x space sec squared straight x over denominator left parenthesis 1 plus tan space straight x right parenthesis squared end fraction

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