Calculating Derivatives

Two kinds of formulas, specific (for specific functions) and general. Both are needed to solve polynomials.

This only works with radians!
Loose summary of derivation:

• \(\lim_{x\rightarrow0} \frac{sin(x+\Delta x) - f(x)}{\Delta x}\)
  
• Use trigonometric identities to expand the \(sin(x+\Delta x)\) term.
  \(sin(a+b) = sin(a)cos(b) + cos(a)+sin(b)\).
  
• Group the terms by plugging in the derivative at \(x=0\) such that the resulting quotients end up cancelling.
  

\(sin(x)\bigg(\frac{cos(\Delta x)-1}{\Delta x}\bigg) + cos(x)\bigg(\frac{sin(\Delta x)}{\Delta x}\bigg)\)
Right side's quotient goes to \(1\), left side goes to \(0\), so the entire equation goes to \(cos(x)\)
The derivatives of sin and cosine at \(x=0\) give all values of the derivative of sine and cosine.

Similar derivation to above which yields \(-sin(x)\).

• Product rule: \((uv)' = u'v + uv'\)
  
• Quotient rule: \((\frac{u}{v})' = \frac{u'v - uv'}{v^2}\)