DIFFERENTIATION RULES & STATIONARY POINTS AND INFLEXIONS

DIFFERENTIATION RULES & MORE

Let $k$ be some constant number (real number).

• $\large y=k$, $\large y'=0$
• $\large y=kx$, $\large y'=k$

Power rule  

• $\large y=x^n$, $\large y'=n\,x^{n-1}$
•  $\large y=k\,x^n$, $\large y'=kn\,x^{n-1}$

Product Rule

• $\large y=uv$,  $\large y'=vu'+uv'$

Quotient Rule

• $\large y=\displaystyle {u\over v}$, 
•  $\large y'=\displaystyle {vu' - uv' \over v^2} $

Chain Rule (for powers of a function)

• $\large y=(x+1)^n$, $\large y'=n(x+1)^{n-1}$
• $\large y=(ax+b)^n$, $\large y'=na(ax+b)^{n-1}$
• $\large y=[f(x)]^n$,  $\large y'=n\,[f(x)]^{n-1}f'(x)$

Stationary Points

• Points where $y'=0$.
• Use $y'$ box to find nature of stationary point (i.e. min, max or horiz inflexion)
• Can also use $y''$ to determine nature of stationary points as long as $y''\ne 0$.

Points of Inflexion.

• $y''$=0 at ALL inflexions.
• For possible inflexions solve $y''=0$
• Use $y''$ box to verify inflexions.
• [Note that there exist points where $y''=0$ but which are NOT inflexions, which is why we must use $y''$ box to check them.]

JH