calc exam 2 (derivatives)

d/dx [log_ax]=

\frac{1}{x\ln(a)}

circle equation

x²+y²=r²

trig equation thing to memorize

sin²y + cos²y = 1

d/dx [arccos x]=

—1 / √(1-x²)

d/dx [arctan x]=

1 / (1+x²)

d/dx [arccsc x]=

—1 / x√(x²-1)

d/dx [arcsec x]=

1 / x√(x²-1)

d/dx [arccot x]=

—1 / (1+x²)

d/dx [arcsin x]=

1 / √(1-x²)

chain rule

(f o g)’ = f’(g(x)) * g’(x)

or can do y=f(u) u=g(x)

quotient rule: d/dx of f(x)/g(x)

(gf’-fg’) / g²

trig limits to memorize

lim_x→0 (sinx)/x = 1

lim_x→0 (cosx -1)/x = 0

sin(a+b)=

= sin(a)cos(b) + cos(a)sin(b)

d/dx [sinx]=

= cosx

d/dx [cosx]=

= —sinx

d/dx [tanx]=

= sec²x

d/dx [cscx]=

= —cscx*cotx

d/dx [secx]=

= secx*tanx

d/dx [cotx]=

= —csc²x

d/dx ( c )=

= 0

d/dx [ f(x) ± g(x) ]=

= d/dx[ f(x) ] ± d/dx[ g(x) ]

d/dx [e^x ]=

= e^x

d/dx [ lnx ]=

= 1/x

d/dx [c*f(x) ]=

= c * f ’(x)

differentiable

well defined slope at the point, can take derivative (ex: NO for absolute value fxn bc of sharp point, or no when slope undefined ie purely vertical)

quote

e^x only thing that can destroy it is itself, -mr incredible of the incredibles movie

product rule: d/dx [fg]=

= f’g + g’f

“take turns holding the mic”

limit definition of derivitive

lim_h→0 [f(a+h)—f(a)] /h

newtons notation

f ‘(x) = (f(x))’

leibnitz notation

df/dx = d/dx [f(x)] = d/dx [f]

sketching derivitive graph and identifing derivative graph

practice more

d/dx [b^x]=

b^x * ln(b)