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 src="data:image/png;base64,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