What is the derivative of sin 2x from first principles?

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justaguide eNotes educator| Certified Educator

The derivative of sin 2x has to be determined from first principles.

For a function f(x) the derivative from first principles is `lim_(h->0)(f(x+h)- f(x))/h`

Using f(x) = sin 2x, the derivative is:

`lim_(h->0)(sin(2*(x+h))- sin 2x)/h`

=> `lim_(h->0)(sin(2x+2h)- sin 2x)/h`

=> `lim_(h->0)(2*cos((2x + 2h + 2x)/2)*sin((2x+2h - 2x)/2))/h`

=> `lim_(h->0)(2*cos (2x+h)*sin h)/h`

=> `lim_(h->0)2*cos (2x+h)*(sin h)/h`

=> `lim_(h->0)2*cos (2x+h)*lim_(h->0)(sin h/h)`

Use `lim_(h->0)(sin h/h) = 1` and substituting h = 0

=> `2*cos 2x`

The derivative of sin 2x is 2*cos 2x