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.

Answered by Laasya S. Maths tutor

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