Differential calculus Flashcards
(30 cards)
What is the derivative of cos?
-sin
What is the derivative of tan?
sec
lim_ x to 0
equation:
x^6 + 9x / x
this equation is undefined (since x is 0), so how do you solve this?
factor x^6 + 9x
becomes
x(x^5 +9) / x
cancel
x^5 + 9
sub and thats your answer.
how do you find m_tan?
you first must find m_PQ which is
f(x+h) - f(x) / h
so if you have y=4x^2
it becomes
4(x+h)^2 - 4x^2 / h
simplify
8x + 4h
m_tan is when h is 0
so m_tan is 8x
then sub it in whatever x is and that’s your slope.
f(x+h) - f(x) / h
what does that also includes
if question is 2x^2 + 2x
it becomes
2(x+h)^2 + 2(x+h) - (2x^2 +2x) / h
how do you find the average rate of change?
just like how you find the gradient
y_2 - y_1 / x_2 - x_1
how do you find the instanteous rate?
f(x+h) - f(x) / h
sometimes you dont need to do that, sometimes you just do the power rule derivative then sub it.
how do you find m_PQ?
same step for finding m_tan but don’t make h = 0
h = x_2 - x_1
then sub that answer to the final expression after
f(x+h) - f(x) / h
h is not 0 when finding m_PQ
how do you find the points where a tangent line to the curve has a m_tan that is defined?
do f(x+h) - f(x) / h to the equation
then however answer you have do this
m_tan = f(x) (simplified)
= x_1
then do
f(x_1)^2 = x_2
then to becomes x_1 , x_2
eg
y = 7x^2, m_tan = 28
simplified 7x^2 is 14x
14x = 28
x = 2
14(2)^2 = 28
so points are
(2,28)
how do you find the slope of the perpendicular line?
usual m_tan
but the final answer must be -1/x
This is because two perpendicular lines, if their slopes are both negative reciprocals of each other, it results in -1.
eg 7-4x^2, x = -1
simplified
-8x
-8(-1) = 8
knowing the rule
-1/8
what is the difference between expression and result?
expression is just the formula
result is the answer
expression can also be the answer considering there’s no variables when simplified.
what does a =?
m_tan expression
when the function
f(x+h) - f(x) gets messy, what do you do?
start with f(x+h) - f(x)
result of that
then do result /h
simplfied whatever.
what is the derivative of a sum or minus function like
9x^4 + 6x^3 - x + 1/x^2
ez
just do each part
for 1/x^2, turn it into x^-2
-2x^-3 (since power is minus 1)
then do -2/x^3
when do you use the limit or rules?
limit if the foundation essence
if a question seems complex like is a polynomial, do rules.
if the questions says f’(x)
do limit rule
for what point on the curve of y = 9x^2 + 5x is the slope of a tangent line equal to 95?
so find the derivative of 9x^2 + 5x
becomes 18x + 5
do 95 = 18x+5
answer is 5
now go back and do
9x^2 + 5x and sub 5
is 250
overall
(5,250)
if the questions looks like
(x)(y)
how do you find the derivative?
product rule or even better
simplify it then find the derivative of each using power rule.
if the question looks like x/y
what do you do?
quotient rule.
x sqrt(x), what can this been simplfied too
x^3/2
derivative
3/2 x ^1/2
what should you beware of?
sometimes when solving using rules, you would have to use rules to find the derivatives of the numbers further
eg
1/ (2x+5)^7
quotient rule
but (2x+5)^7, to find its derivative, you need to do
chain rule
so this can just become
(2x+5)^-7
do chain rule and boom
x sqrt(x)^5, what does this becomes in derivative?
1/5 x ^-4/5
becomes 1/5x^4/5
what does d^2y / dx^2 means or y “
or even more numbers or dashes
derive more
so for example
y = 10x^3 +9x^2
y’ = 30x^2 + 18x
y ‘’ = 60x + 18
y’’’ = 60
y ‘’’’ = 0
and continue on and on until it reaches 0.
what is the derivaive of 6pi^2
usual 12pi
if the first derivative is the displacement, what is the second derivative?
acceleration
