2.3.1: Resources and Key Concepts — Read

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Resources

Theorem: The Derivatives of \(\sin(x)\) and \(\cos(x)\)
- \(\frac{d}{dx}(\sin(x)) = \cos(x)\)
- \(\frac{d}{dx}(\cos(x)) = -\sin(x)\)
Theorem: Derivatives of \(\tan(x), \cot(x), \sec(x)\), and \(\csc(x)\)
- \(\frac{d}{dx}(\tan(x)) = \sec^2(x)\)
- \(\frac{d}{dx}(\cot(x)) = -\csc^2(x)\)
- \(\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)\)
- \(\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)\)

Forgetting Radian Measure: The proofs for the derivatives of trigonometric functions (especially \(\sin x\) and \(\cos x\), which rely on \(\displaystyle \lim_{h \to 0} \frac{\sin h}{h} = 1\) and \(\displaystyle \lim_{h \to 0} \frac{\cos h - 1}{h} = 0\)) assume radian measure. Working in degrees would require significant adjustments.
Sign Errors in Derivatives: Forgetting the negative signs in the derivatives of cosine, cosecant, and cotangent.
Oversimplification or Lack of Simplification: Trigonometric expressions can often be simplified using identities. While not always leading to a "simpler" final form, it's advisable to check for possible simplifications after differentiation (see end of Example 7).

This page titled 2.3.1: Resources and Key Concepts was last modified on Mon, 22 Sep 2025 20:52:01 GMT and is shared under a CC BY-SA license and was authored, remixed, and/or curated by Roy Simpson.