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 <img 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