2.1-2.5 quiz Flashcards
sin x ‘
cos x
cos x’
-sin x
tan x’
sec^2 x
cot x
-csc^2 x
sec x’
sec x(tan x)
csc x’
-csc x(cot x)
product rule
h(x) = f(x)(g(x))
h’(x) = f x g’ + g x f’
quotient rule
h(x) = f(x)/g(x)
h’(x) = g x f’ - f x g’ / g^2
chain rule
h(x) = f(g(x))
think like peeling an onion
h’(x) = f’(g(x)) x g’(x)
sin 2x
2sinxcosx
consider the curve : formula
show that dy/dx equals formula
basically dy/dx is the formula of the curve
what is differentiation
where the slope equals the tangent line
the process of finding a derivative is called
differentiation
what does differentiability imply
continuity
the derivative doesn’t exist at
discontinuities
cusps
when the tangent becomes vertical(straight up and down line part)
power rule
d/dx[x^n] =
nx^n-1
power rule on constants
d/dx 5
0
find equation of tangent line to graph of f(x) at the point(x,y)
how to solve
find derivative of f(x) to solve for the slope at any point on the function
then plug in (x,y) to find the slope for that particular point
implicit differentiation procedure
differentiate each term with respect to x (dy/dx)
nivel all terms contains dy/dx to the left side of the equation and all other terms to the right side
factor out dy/dx on left side of equation
solve for dy/dx
product vs chain tule for trig functions:
if the angle theta is just x it is product rule; if it is 2x however or (2x-1) whether it has parentheses or not it is chain rule
what happens to the “y” with implicit differentiation
it becomes dy/dx
with respect to x
what happens to x in implicit differentiation
nothing with respect to x becomes 1 so the x is left alone
L’hospitsl rule
How to wrote AP TESTA ND FRQ
1) show f(x) and g(x) are continuos and differentiable
(Write f(x) and g(x) separately!)
2) show lim f(x) = 0 and lim g(x) = 0
x->a x-> a
3) using L’Hospitals rule
lim f(x)/ g(x) = lim f’(x)/g’(x) plug in number
x->a x-> a
L’hospital rule idea
If in indeterminate form find derivative of og equation and replug in pts, if still indeterminate keep finding derivatives
finding second derivative
determine what is f(x) and g(x) and f’(x) and g’(x) [still use implicit differentiation plug in dy/dx]
then solve
for the dy/dx (it equals previous derivative
D’)
derivative of “xy”
use product rule
(xdy/dx + y1)
keep subtraction sign outside of parentheses and distribute once finished differentiating
constant on right side of equal sign
don’t forget to differentiate it!!
equals 0
how to write second differentiation in (dy/dx) format
d^2y/dx^2
??
when solving any typa derivative problem
first identify and write down f(x), f’(x), g(x), g’(x)
then identify what rule to use and write that down
write down formula for rule too if needed
differentiation of 4xy
group (4x) as f(x) and (y) as g(x)
becuase 4 is only multiplied to the y(not distributed to the whole thing)
when doing implicit differentiation only put derivatives on y when
differentiating for y
for example in product rule where f(x)*g’(x) if f(x) = y DONT put dy/dx becuase u aren’t supposed to find the derivative of f(x)
differentiating is ONLY used when u need to find the derivative of THAT PARTICULAR variable
is cos^2(x) composite
yes
inner function: cos(x)
outer function: x^2
derivative of sec(4x)
sec(4x)tan(4x)*4
trig function inside of trig function means how many parts of an onion
3 layers of onion
lim f(x+h) -f(x)/ x-h x->h
identify
derivative of f(x) @ any point
derivative of f(x) @ any point is the derivative upper right part of the fraction
f’(x)
instantaneous rate of change
find derivative
then plug in point
average rate of change
f(b)-f(a)/ b-a
