How do I differentiate sin^2(x)?

To differentiate sin^2(x) we must use the 'Chain Rule'. This is because we have a function of a function. We let y=sin^2(x). Then we write, 'let u=sin(x)'. This is our initial function, and we can see now that using this new notation, sin^2(x) is simply u^2. So we now write y=u^2, as this is equivalent to y=sin^2(x). To find dy/dx, we need to apply the chain rule. This states that dy/dx=dy/du x du/dx. To find dy/du we differentiate y with respect to u. Since, y=u^2, we have that dy/du=2u. To find du/dx we differentiate u with respect to x. We have that u=sin(x), so differentiating u with respect to x we have that du/dx=cos(x). Now we simply substitute the values of dy/du and du/dx into the chain rule so that we can obtain a value for dy/dx. We have that dy/dx = dy/du x du/dx, so dy/dx = 2ucos(x). However, we need the final differentiated answer to be in terms of x, as there are no 'u's in the initial expression 'sin^2(x)'. So, since u = sin(x), we subsitute in 'sin(x)' where the letter u appears in our answer for dy/dx. Therefore, instead of writing dy/dx = 2ucos(x), we write dy/dx=2sin(x)cos(x). Now we have successfully differentiated sin^2(x) with respect to x, and have written our answer correctly in terms of x. 

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