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What is the derivative of f(x) = sin(x^2)?

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Question: What is the derivative of f(x) = sin(x^2)?

Options:

  1. 2x cos(x^2)
  2. cos(x^2)
  3. 2x sin(x^2)
  4. sin(x^2)

Correct Answer: 2x cos(x^2)

Solution: 

Using the chain rule, f\'(x) = cos(x^2) * 2x = 2x cos(x^2).

What is the derivative of f(x) = sin(x^2)?

Practice Questions

Q1
What is the derivative of f(x) = sin(x^2)?
  1. 2x cos(x^2)
  2. cos(x^2)
  3. 2x sin(x^2)
  4. sin(x^2)

Questions & Step-by-Step Solutions

What is the derivative of f(x) = sin(x^2)?
Correct Answer: 2x cos(x^2)
  • Step 1: Identify the function we want to differentiate, which is f(x) = sin(x^2).
  • Step 2: Recognize that this is a composite function, where the outer function is sin(u) and the inner function is u = x^2.
  • Step 3: Apply the chain rule, which states that the derivative of sin(u) is cos(u) multiplied by the derivative of u.
  • Step 4: Find the derivative of the inner function u = x^2. The derivative is u' = 2x.
  • Step 5: Now, apply the chain rule: f'(x) = cos(x^2) * (derivative of x^2).
  • Step 6: Substitute the derivative of the inner function: f'(x) = cos(x^2) * 2x.
  • Step 7: Simplify the expression to get the final answer: f'(x) = 2x cos(x^2).
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