How do I differentiate sin 2 ⁡ x?I thought that because this is true: sin...

chiodaiokaC

chiodaiokaC

Answered

2022-11-26

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?

Answer & Explanation

Alayna Phillips

Alayna Phillips

Expert

2022-11-27Added 8 answers

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't use the power rule because it's not just an x term being raised to a constant power, it'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

ramirezherePYM

Expert

2022-11-28Added 1 answers

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

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