Question

Derivatives of sin square x

Answer 1

2sin x cos x or sin(2x)

Answer 2

To find the derivative of $\sin^2 x$, we use the chain rule.

Let $y = \sin^2 x$.
This can be rewritten as $y = (\sin x)^2$.

We apply the chain rule, which states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

1. Identify the outer function and the inner function:
   Let $u = \sin x$ (inner function).
   Then $y = u^2$ (outer function).

2. Differentiate the outer function with respect to $u$:
   $\frac{dy}{du} = \frac{d}{du}(u^2) = 2u$.

3. Differentiate the inner function with respect to $x$:
   $\frac{du}{dx} = \frac{d}{dx}(\sin x) = \cos x$.

4. Apply the chain rule:
   $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
   Substitute the expressions from steps 2 and 3:
   $\frac{dy}{dx} = (2u) \cdot (\cos x)$

5. Substitute back $u = \sin x$:
   $\frac{dy}{dx} = 2(\sin x)(\cos x)$

This result can also be expressed using the trigonometric identity $\sin(2x) = 2\sin x \cos x$.
Therefore, $\frac{dy}{dx} = \sin(2x)$.