Limits and Derivatives

Limits and Derivatives

Question

Find the derivatives of the following functions :

space space space space space space space space fraction numerator sec space straight x space minus 1 over denominator sec space straight x space plus 1 end fraction

Answer

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

  space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis sec space straight x plus 1 right parenthesis left parenthesis sec space straight x space tan space straight x minus 0 right parenthesis minus left parenthesis sec space straight x minus 1 right parenthesis left parenthesis sec space straight x space tan space straight x plus 0 right parenthesis over denominator left parenthesis sec space straight x plus 1 right parenthesis squared end fraction
space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator sec squared straight x space tan space straight x space plus sec space straight x space tan space straight x minus sec squared space xtan space straight x space plus space sec space straight x space tan space straight x over denominator left parenthesis sec space straight x space plus 1 right parenthesis squared end fraction equals fraction numerator 2 space sec space straight x space tan space straight x over denominator left parenthesis sec space straight x space plus 1 right parenthesis squared end fraction

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