How do I differentiate sin 2 ⁡ x?I thought that because this is true: sin 2 ⁡ x = ( sin ⁡ x ) 2 ,I could differentiate the expression like this: d d x sin 2 ⁡ x = 2 cos ⁡ x .But I am supposed to get sin ⁡ ( 2 x ) or 2 sin ⁡ x cos ⁡ x .Why am I wrong?

Question

How do I differentiate       sin    2    ⁡  x?I thought that because this is true:      sin    2    ⁡  x  =  (  sin  ⁡  x      )    2    ,I could differentiate the expression like this:      d          d      x            sin    2    ⁡  x  =  2  cos  ⁡  x  .But I am supposed to get  sin  ⁡  (  2  x  )    or    2  sin  ⁡  x  cos  ⁡  x  .Why am I wrong?

Alayna Phillips · Accepted Answer

You can only use the rule:       d          d      x            x    n    =  n      x          n      −      1       if x is just x and not a function. In the case for       sin    2    ⁡      x   then as you write it in the form   (  sin  ⁡      x        )    2   you can see that we can&#039;t use the power rule because it&#039;s not just an x term being raised to a constant power, it&#039;s a function being raised to a constant power. To differentiate       sin    2    ⁡      x   one must use the Chain Rule because   sin  ⁡      x   is a function of x within another function (the function that is squaring   sin  ⁡      x  )       x    2  . The chain rule is:       d          d      x        f  (  g  (  x  )  )  =      g    ′    (  x  )  ×  (      f    ′    (  g  (  x  )  ) so you can apply that rule for this case with   f  (  x  )  =      x    2   and   g  (  x  )  =  sin  ⁡      x

ramirezherePYM · Accepted Answer

You are using the chain rule incorrectly:      d          d      x        f  (  g  (  x  )  )  =            d      f              d      x        (  g  (  x  )  )            d      g              d      x        (  x  )Now   f  (  x  )  =      x    2   and   g  (  x  )  =  sin  ⁡  (  x  ), therefore      d          d      x        (  sin  ⁡  (  x  )      )    2    =                              2          sin          ⁡          (          x          )                ⏞                            f        ′            (      g      (      x      )      )        ⋅                              cos          ⁡          (          x          )                ⏞                            g        ′            (      x      )      And to explain your other point of confusion:   sin  ⁡  (  2  x  )  =  2  sin  ⁡  (  x  )  cos  ⁡  (  x  ) is a well-known trigonometric formula

How do I differentiate sin^2 x?

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