Find the derivative of the function: $f\left(x\right)=cos\left(x^2\right)$. Which of the following is correct?

• A. $f\text{'}\left(x\right)=-2xcos\left(x^2\right)$
• B. $f\text{'}\left(x\right)=2xcos\left(x^2\right)$
• C. $f\text{'}\left(x\right)=-sin\left(x^2\right)$
• D. $f\text{'}\left(x\right)=-2xsin\left(x^2\right)$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the function f(x) = cos(x^2) is a composite function. This means we need to apply the chain rule to find its derivative.
2. Step 2: Recall the chain rule formula: If a function is composed as f(g(x)), then its derivative is f'(g(x)) * g'(x). Here, f(x) = cos(u) and u = x^2.
3. Step 3: Differentiate the outer function cos(u) with respect to u. The derivative of cos(u) is -sin(u). So, f'(u) = -sin(u).
4. Step 4: Differentiate the inner function u = x^2 with respect to x. The derivative of x^2 is 2x.
5. Step 5: Combine the results using the chain rule: f'(x) = -sin(x^2) * 2x. Simplify to get f'(x) = -2x * sin(x^2).