How do you differentiate Sin[x^2]?

Question

How do you differentiate Sin[x^2]?
i.e. NOT (Sin[x])^2, but with the square operator actually part of the argument of sine.

Answer 1

Use the chain rule:

(sin(x^2))' = (x^2)' cos(x^2), because the derivative of the sine is the cosine. Since the derivative of x^2 is 2x, it follows that
(sin(x^2))' = 2x cos(x^2)

Answer 2

Sin[x^2]

1) Take derivative of Sin(x^2). (Just ignore the x^2 for now):
cos(x^2)

2) Next, take derivative of x^2 and multiply it with Cos(x^2):

Answer: 2x*cos(x^2)

Answer 3

sin(x)^2

let x^2 = y

2x = dy/dx

now d(sin(x)^2)

=> d(sin y)

=> cos(y) dy/dx

substitute back y = x^2 and dy/dx = 2x

cos(x)^2 *2x

=>2x cos(x^2)