Unit 2 Differentiation Flashcards
Average Rate of Change
F(x) - f(a)
————-
X - a
Average rate of change is the slope of the _________________ line.
Secant
Instantaneous Rate of Change
Lim [f(x+h) - f(x)] / h
h ->0
Instantaneous rate of change is the slope of the ______________ line.
Tangent
Limit Definition of Derivative
F’(x) = lim [(f(x+h) - f(x)] / h
h->0
A ________________ gives the slope of the tangent line
Derivative
Equation of tangent line at x=a
Y - f(a) = f ‘(a) (x - a)
Three way a function is not differential:
1) discontinuity
2) corner/cusp
3) vertical tangent
Definition of differentiability
1) derivative exists for each point in domain; smooth line or curve
2) graph looks like a line if zoomed in (local linearity)
The Power Rule
Derivative of f(x) = x^n
F’(x) = nx^(n-1)
What is true of parallel tangent lines?
Derivatives (slopes) will be equal
Constant Derivative Rule
dy/dx C =
0
Constant Multiple Rule
dy/dx 3x^4 =
3 * 4x^3 = 12x^3
Sum/Difference Derivative Rule
dy/dx (3x^5 + 6x^2 - 7x - 9) =
15x^4 + 12x - 7
Horizontal tangent lines have a slope of ___________ because it’s an max/min point.
0
_______________________ go through the same point as tangent line, but is perpendicular to tangent line.
Normal lines
Two things to check for differentiability of piecewise functions
1) continuity - to make sure the function is continuous
2) equal derivatives - to make sure there is not a corner/cusp in the continuous function
Derivative of sin x
Cos x
Derivative of cos x
- sin x
Derivative of a^x
a^x lna
Derivative of e^x
e^x
Derivative of log base a of x
(1/x)(1/lna)
Derivative of ln x
1/x
Product Rule
If h(x)=f(x) g(x), then h’(x) =
h’(x) = f’(x) g(x) + f(x) g’(x)
Quotient Rule
If h(x)=f(x)/g(x), then h’(x) =
h’(x) = [f’(x) g(x) - f(x) g’(x)] / [g(x)]^2
Derivative of tan x
sec^2 x
Derivative of cot x
-csc^2 x
Derivative of sec x
(sec x)(tan x)
Derivative of csc x
(-csc x)(cot x)
