How do I differentiate sin^2(x)?
To differentiate this, we must use the Chain Rule. This is because we have two functions multiplied by each other:
(x) sin(x).
We use the substitution u = sin(x). This is our initial function, and we can see now that using this new notation, y = sin2(x) is simply y = u2.
To find:
We need to apply the chain rule. This states that:
To find:
We differentiate y with respect to u. Since y= u2, we have that:
To find:
We differentiate u with respect to x. We have that:
So, differentiating u with respect to x, we have that:
Now, we simply substitute the values of:
and 
into the chain rule, so that we can obtain a value for 
.
We have that:
However, we need the final differentiated answer to be in terms of x, as there are no ‘u's in the initial expression y = sin2(x).
So, since u = sin(x), we substitute in sin(x) where the letter u appears in our answer for:
Therefore, instead of writing:
We write:
Now we have successfully differentiated y = sin2(x) with respect to x and we have written our answer correctly in terms of x.
