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Calculating Higher-Order Derivatives

AP Calculus AB· difficulty 4/5

If f(x)=sin⁡xf(x) = \sin x, then f(2026)(0)=f^{(2026)}(0) =

  • A

    00

    check_circle
  • B

    20262026

  • C

    −1-1

  • D

    11

Explanation

Pattern: f(0)=sin⁡,f(1)=cos⁡,f(2)=−sin⁡,f(3)=−cos⁡f^{(0)}=\sin, f^{(1)}=\cos, f^{(2)}=-\sin, f^{(3)}=-\cos, period 4. 2026mod  4=22026 \mod 4 = 2, so f(2026)=−sin⁡xf^{(2026)}=-\sin x. At 0: 0.

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