Derivative
Differentiability&Continuity
Product Rule
Quotient Rule
Trig Derivatives
100
The derivative calculates the ________.
Slope
100

What are the three conditions that must exist for a function to be considered continuous?

1. f (a) is defined

2. The limit exist(left hand limit equal the right hand limit)

3.f(a) = limit

100
Define the Product Rule using f(x)g(x)
f′(x)g(x)+ g′(x)f(x)
100
Define the Quotient Rule using f(x)/g(x)
[g(x)f′(x) - f(x)g′(x)]/ (g(x))²
100
d/dx cosx=
-sinx
200
d/dx 5 =
0
200

List the four cases that a function is not differentiable

1. cusp

2. corner

3. vertical tangent

4. discontinuous

200
f(x)=x²sinx, what is f′(x)?
2xsinx+ x²cosx
200
Differentiate y= 2/(x+1)
y′ = -2/ (x+1)²
200
Differentiate y=tan(x)
y′ =sec²(x)
300
d/dx x² =
2x
300

What value of k will make this function continuous?


            x^2 - 1, if x<3

f(x0=

         2kx if x>= 3

        

k = 4/3

300
Differentiate y=x³lnx
y′ =x²(1+3lnx)
300
Differentiate y= (1+lnx) / (x²-lnx)
y′= [(1/x)-x-2xlnx] / (x²-lnx)²
300
Differentiate y=csc(x)
y′ =-csc(x)cot(x)
400
d/dx 3x²-x+3 =
6x-1
400

Determine if the function is differentiable

f(x)= l x - 3 l

No

400
Differentiate y=e^-x²cos2x
y′ =−2xe^(−x²) cos2x−2e^(−x²)sin2x
400
f(x)= (x²-1)³/ x²+1, what is f′(x)?
f′(x)= [4x(x²-1)²(x²+2)] / (x²+1)²
400
d/dx sin(2x)
2cos(2x)
500

Average velocity: slope of secant lines::instantaneous velocity:__________________________

slope of tangent line

500

True or false AND why

Continuity implies differentiability

False because a function can be continuous and not differentiable. The left and right hand limit of the derivative won't be equal 

500
Differentiate y=x²sin³(5x)
y′ =xsin²(5x)[15xcos(5x)+2sin(5x)]
500
Differentiate y= (x³lnx)/(x+2)
y′ = [x²(2xlnx+6lnx+x+2)]/ (x+2)²
500

d/dx 14 tan(x) cos(x) + 10csc( x) =

14cosx - 10 cscx cotx OR 14sec^2x cosx-14tanxsinx-10 cscx cotx

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