Question

sin x square differentiation

Answer 1

2x * cos(x^2)

Answer 2

The problem asks for the differentiation of "sin x square". This is interpreted as differentiating the function $y = \sin(x^2)$.

To differentiate $y = \sin(x^2)$, we use the chain rule. The chain rule states that if $y = f(g(x))$, then its derivative with respect to $x$ is given by: $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

In this case:

1. Identify the outer function $f(u)$ and the inner function $g(x)$. Let $u = x^2$. Then the function becomes $y = \sin(u)$. So, $f(u) = \sin(u)$ and $g(x) = x^2$.

2. Differentiate the outer function $f(u)$ with respect to $u$: $\frac{df}{du} = \frac{d}{du}(\sin(u)) = \cos(u)$

3. Differentiate the inner function $g(x)$ with respect to $x$: $\frac{dg}{dx} = \frac{d}{dx}(x^2) = 2x$

4. Apply the chain rule by multiplying the results from steps 2 and 3, and substitute $u$ back with $x^2$: $\frac{dy}{dx} = \frac{df}{du} \cdot \frac{dg}{dx} = \cos(u) \cdot 2x$ Substitute $u = x^2$: $\frac{dy}{dx} = \cos(x^2) \cdot 2x$ Rearranging the terms for better readability: $\frac{dy}{dx} = 2x \cos(x^2)$

The differentiation of $\sin(x^2)$ is $2x \cos(x^2)$.

The other possible interpretation, though less likely from the phrasing "sin x square", is $(\sin x)^2$. If $y = (\sin x)^2$, then: Let $u = \sin x$. Then $y = u^2$. $\frac{dy}{du} = \frac{d}{du}(u^2) = 2u$ $\frac{du}{dx} = \frac{d}{dx}(\sin x) = \cos x$ Applying the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot \cos x$ Substitute $u = \sin x$: $\frac{dy}{dx} = 2 \sin x \cos x$ Using the double angle identity $\sin(2x) = 2 \sin x \cos x$: $\frac{dy}{dx} = \sin(2x)$ However, the phrasing "sin x square" most commonly refers to $\sin(x^2)$.

Solution:

To differentiate $\sin(x^2)$, use the chain rule. Let $y = \sin(u)$ where $u = x^2$. The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$. The derivative of $x^2$ with respect to $x$ is $2x$. Multiplying these derivatives: $\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x = 2x \cos(x^2)$.

Answer:

The differentiation of sin x square (interpreted as $\sin(x^2)$) is $2x \cos(x^2)$.