Recording-2026-01-26T21:10:23.542Z

Derivatives and Antiderivatives

• Derivative of Sine Function:

  • The derivative of the function $ext{sin}(x)$ is $ext{cos}(x)$.

• General Antiderivative of Cosine Function:

  • The most general antiderivative of $ext{cos}(x)$ is $ext{sin}(x) + c$, where $c$ is a constant.

  • **Verification:

  • Derivative Check:**

  • To confirm, take the derivative of $ext{sin}(x) + c$:

    • The derivative of a constant is zero, hence:

    • $rac{d}{dx}( ext{sin}(x) + c) = ext{cos}(x)$.

• Symbolism of Antiderivatives:

  • The notation for the indefinite integral $extstyle rac{d}{dx} F(x) = f(x)$ implies:

  • $F(x)$ is the antiderivative of $f(x)$, denoted as $ext{F}(x) + c$.

• Multiple Antiderivatives:

  • A function can have infinitely many antiderivatives differing by a constant.

  • For instance, the antiderivative of $ext{cos}(x)$ using constants will be:
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