Test 3 AP Calc AB

Derivative of y=sinx

cosx

Derivative of y=cosx

-sinx

Derivative of y=tanx

sec²x

Derivative of y=cscx

-cscxcotx

Derivative of y=secx

secxtanx

Derivative of y=cotx

-csc²x

Chain Rule

y= f(g(x))

y’ = f’(g(x)) = g'(x)

I'm mplicit Differentiation

Derivatives with a mixture of x and y

To solve dy/dx

1. Move the values without dy/dx to other side

2. Factor out what does and divide it moving it to other side

What to remember when using product or quotient rule with a - in front

The - goes to all parts of it

If a problem has two y’s, you get two…

y’

For an e^x chain rule, keep the exponent then multiply by…

The derivative of the exponent since that's the inside

Limits

As x gets dealt close to “a”, y gets really close to “L”

When approaching a point in a limit, does it matter if you hit it or not?

No, it does not matter → a hole can have a limit!

Why does a limit not exist?

Because y’s do not match up

Sum Law

lim x→a[f(x)+g(x)] = limf(x) x→a + limg(x) x→a

The limit of a sum is the sum of the limits.

Difference Law

lim x→a[f(x)-g(x)] = limf(x) x→a - limg(x) x→a

The limit of a difference is the difference of limits.

Constant Multiple Law

lim x→a[cf(x)] = climf(x) x→a

The limit of a constant times a function is the constant times the limit of the function.

Product Law

lim x→a[f(x)•g(x)] = limf(x) x→a • limg(x) x→a

The limit of a product is the product of the limits.

Quotient Law

lim x→a[f(x)/g(x)] = limf(x) x→a/limg(x) x→a

The limit of a quotient is the quotient of the limits.

For a VA, to figure out whether it's -/+ infinity, you…

Plug in a number a little higher and a little lower than the bottom number

Make sure when doing limits you continue to…

Write lim x→#

If limf(x)= L x→±∞

Then y=L is a horizontal asymptote

If top heavy in a limit…

Compare the powers with the coefficients involved!

For limits equations…

Always plug in!

#/0 =

VA

0/0 =

Hole

If asking to check whether the limit exists,

set the first and second expression equal + plug in the limit. Then, do the same and say the middle and bottom to each other and plug in next limit. Now you have a system of equations: graph, substitute, or eliminate.

logb(1)=

0

logb(b) =

1

Continuous

Never lift the pencil

A fraction f(x) is continuous at x=a if:

1) f(a) must exist. There is a y value there.

2) limf(x) x→ a must exist.

3) f(a)=limf(x) x→ a must equal each other

3 Types of Discontinuity

1. Removable = hole

2. Jump → physical disconnect

3. Infinite → VA

Derivative also means…

Slope!

Definition of a Derivative

f’(x) = lim h→0 f(x+h)-f(x)/h

Second definition fo derivative

f’(a) = lim x→a f(x)-f(a)/x-a

The derivative does not exist at x=a if…

1. f(x) is discontinuous at x=a

2. f(x) has a corner at x=a

3. f(x) has vertical tangent at x=a

What's something Krausman always tells us

Don't forget to distribute the negative!

Power Rule

y=ax^n

y’=(a)(n)x^(n-1)

Exponent Rule

y=e^x

y=e^x

Log Rule

y=lnx

y’=1/x

A Quadratic has a… derivative

Linear

When do you use chain rule?

When you have a composite function (a function inside another function), and you do not use it when you have a basic, single-layer function.

Common ones:

• Power inside a power

• Trig without an x on the outside

• Exponential or logarithmic functiona

When doing the chain rule, check for…

Any product or quotient rule stuff! If you see something like this…

t(sin(πt)), you're doing t•sin(πt) meaning you are doing product rule and chain rule!

d²y/dx² just means…

Second Derivative

The derivative of position is…

Velocity

The derivative of velocity is…

Acceleration

Acceleration measurement

m/s²

Velocity measurement…

m/s

y=x²

y=√x

y=|x|

y=lnx

y=sinx

y=x³

y=³√x

y=e^x

y=1/x

y=cosx

When finding the limit, you only use HA when

Limit is going to infinity or negative infinity

Difference of cubes: SOAP

1st is same sign, 2nd is opposite sign, last is always positive

Limit going to an exponent that is rational…

Divide by the highest degree in exponent, and then find the number to the exponent →

Ex: 2^(3x+4/x+2)

= 2³ = 8