Liza G's Final Math Project

Derivatives

100

What's the Derivative?

f(x)=sin

f'(x) = cos(x)

200

What's the Derivative?

f(x)=cos(x)

f'(x) = − sin(x)

300

What's the Derivative?

f(x)=sec(x)

f'(x) = sec(x) tan(x)

400

What's the Derivative?

f(x)=tan(x)

f'(x) = sec²(x)

500

What's the Derivative?

f(x)=cot(x)

f'(x) = − csc²(x)

Chain Rule

100

y=(2x²+x)²⁰⁰

^{What would the inner function be?}

I= 2x²+x

200

y= 1/(x³-2x²+7)⁴

What is the outer function for this equation?

O= 1/(x³-2x²+7)⁴

300

I=x³

What is the derivative (aka I' ) of x³

I'=3x²   

400

I= x²+1

What is the derivative of I. 

I' = 2x

500

f(x)=sin⁵2x

How many inner functions will there be?

There will be 2 inner functions 

I₁=sin2x

I₂=2x

Different Rules

100

What is this rule called?

(f'g-g'f)/g²

Quotient Rule

200

What is this rule called?

h(x) =f (g(x))g(x)

Chain Rule 

300

What is this rule called?

f(x) = c then f 0 (x) = 0

Constant Rule

400

What is this rule called?

(xⁿ)¹=nx^n-1

Power Rule

500

What is this rule called?

g(x) = c · f(x) then g ·(x) = c · f 0 (x)

Constant Multiple Rule

Critical Numbers

100

What is the critical number of this function?

3(x-1)(x-3)

x=1, x=3

200

What is the critical number of this function?

(1/4)-x=0

x= 1/4

300

What are the critical numbers of this function?

f'(x)= 3x²+12x-96

3(x+8)(x-4)=0

x=-8, x=4

400

What are the critical numbers of this function?

f'(x)=15x⁴-15x²

15x²(x-1)(x+1)=0

x=0, x=1, x=-1

500

What are the critical numbers of this function?

f'(x)= 4x³-16x

4x(x-4)(x+4)=0

x=0, x=4, x=-4

Misc.

100

When a line lies BELOW tangent lines it is which concavity?

Concave Downward

200

When a line lies ABOVE tangent lines it is which concavity?

Concave upward

300

When a line is increasing on a graph what is the equation?

f'(x)>0

400

What is the derivative equation for decreasing on a graph?

f'(x)<0

500

When a line on a graph is constant what is the equation?

f'(x)=0