Question

If $f(x) = sin(x^2)$, what is $f'(x)$?

• A. $2x \cos(x^2)$
• B. $sin(x^2) \cdot cos(x)$
• C. $cos(x^2)$
• D. $2x sin(x^2)$

Answer 1

Answer: (A)

$2x \cos(x^2)$

Answer 2

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

Let $y = f(x) = \sin(x^2)$. We can consider this as a composite function where $y = \sin(u)$ and $u = x^2$.

According to the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

First, find the derivative of $y$ with respect to $u$: $\frac{dy}{du} = \frac{d}{du}(\sin(u)) = \cos(u)$

Next, find the derivative of $u$ with respect to $x$: $\frac{du}{dx} = \frac{d}{dx}(x^2) = 2x$

Now, substitute $u = x^2$ back into $\cos(u)$ and multiply the two derivatives: $f'(x) = \cos(x^2) \cdot (2x)$ $f'(x) = 2x \cos(x^2)$