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Derivatives of Trig Functions quiz #1

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  • What is the derivative of f(x) = x^2 sin(x)?
    Using the product rule, the derivative is f'(x) = x^2 cos(x) + 2x sin(x).
  • How can you quickly find the 37th derivative of sin(x) without differentiating 37 times?
    Divide 37 by 4; the remainder is 1, so the 37th derivative is the same as the first derivative: cos(x).
  • What is the derivative of f(x) = 4x sec(x)?
    Using the product rule, the derivative is f'(x) = 4 sec(x) (x tan(x) + 1).
  • What is the derivative of sin(x) with respect to x?
    The derivative of sin(x) is cos(x).
  • How do you find the nth derivative of sin(x) or cos(x) quickly?
    Divide n by 4 and use the remainder to determine which derivative to take; every 4th derivative returns to the original function.
  • What is the derivative of tan(x) and how can it be derived?
    The derivative of tan(x) is sec^2(x), which can be derived using the quotient rule on sin(x)/cos(x).
  • What is the derivative of cot(x) and what pattern do cofunction derivatives follow?
    The derivative of cot(x) is -csc^2(x); all cofunction derivatives (like cosine, cotangent, cosecant) are negative.
  • How do you apply the product rule to find the derivative of f(x) = x^2 sin(x)?
    Use the product rule: f'(x) = x^2 cos(x) + 2x sin(x).
  • How can you quickly find the 37th derivative of sin(x) without differentiating 37 times?
    Divide 37 by 4; the remainder is 1, so the 37th derivative is the same as the first derivative: cos(x).
  • What is the derivative of f(x) = 4x sec(x) using the product rule?
    Using the product rule, the derivative is f'(x) = 4 sec(x) (x tan(x) + 1).