How to find the derivative of y = ln | sec x + tan x | ?

Question

How to find the derivative of       y    =    ln          |              sec        x            +              tan        x            |      ?

Kayden Yates · Accepted Answer

y    ′         =               sec      x        .  Solution: We&#039;ll use the Chain Rule to accomplish this.            Recall:                                y         =         ln          |      x      |                ⇒            y    ′         =               1      x        .  Therefore, by the Chain Rule:                    y         =         ln          |              f                  (          x          )                    |                ⇒            y    ′         =               [              1                  f                      (            x            )                              ]        ⋅    f    ′          (      x      )             =                       f        ′                  (          x          )                            f                  (          x          )                      .                ∴                    y         =         ln          |              f                  (          x          )                    |                ⇒            y    ′         =                       f        ′                  (          x          )                            f                  (          x          )                      .  Thus, in our example:                                              y         =         ln          |              sec        x            +              tan        x            |        ;                                                       f                      (            x            )                                   =                           sec          x                +                  tan          x                                                                y    ′         =                                 (                      sec            x                    +                      tan            x                    )                ′                              sec          x                +                  tan          x                      ;                                        =                                           f            ′                          (              x              )                                            f                          (              x              )                                                                                                   =                                 (                      sec            x                    )                ′        +                  (                      tan            x                    )                ′                              sec          x                +                  tan          x                      ;                                                           =                                 sec          x                          tan          x                +                              sec            2                    x                                      sec          x                +                  tan          x                      ;                                                           =                                 sec          x                          (                      tan            x                    +                      sec            x                    )                                      sec          x                +                  tan          x                      ;                                                           =                                 sec          x                          (                      sec            x                    +                      tan            x                    )                                      sec          x                +                  tan          x                      ;                                                           =                                 sec          x                                                                    (                                  sec                  x                                +                                  tan                  x                                )                                                                                                    (                              sec                x                            +                              tan                x                            )                                            ;  #                                                         =               sec      x        .  Thus, we have now:                                                                                            y    ′         =               sec      x        .  Summarizing:                                      y         =         ln          |              sec        x            +              tan        x            |                ⇒            y    ′         =               sec      x        .

How to find the derivative of y = ln | sec x + tan x...

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