find all possible values of rank(A) as a varies. a 2 13 3 22 1 aA 1 2

Derivatives of Trig Functions: In general, sinx)=cosx ' In general, cosx )=−sinx ' 2 1 In general, tanx )=sec x= 2 (you can derive this cos x using the trig and quotient rules) In general, secx )=tanx∗secx (you can derive this using tri and quotient rules) The Chain Rule: If f is differentiable at g(x)'and g is differentiable at x, then [f ∘g )x ]= f g( (x))g '(x) ' ' ' ' [(f ∘g∘h )(x)]=f g(h( x ))( h (x))h (x) Example: x f (x)=2 f(x = ln2 Remember x 2=e 2 = (eln) =e xln2 xln2 Inner: () Outer: e (Inner)’ = ln2 () (Outer)’ = ' e ' f(x = (2x)= (exln)=e ∗ln2=e xlnln2=2 ln2 ' In general: xlna )=lna In general: a x)=a lnaif a>0 Implicit Differentiation: Know how to solve for y’ when it an equation is not solved for y 2 methods: o 1. You can solve the equation for y and then differentiate o 2. You can differentiate both sides of the equation and isolate y’ (called implicit differentiation). Use the chain rule to differentiate terms with y