Unit 2- Differentiation: Definition & Fundamental Properties Flashcards
Avg. RoC on interval [a,b]
f(b) - f(a) / b-a
If the x-value of point a = a and x-value of point b = a +h, then what will be the formula used for avg. RoC
f(a+h) - f(a) / h
The slope of the secant line can be modeled algebraically by
the avg. RoC formula
The slope of the secant line can be modeled algebraically by
taking the lim of the RoC formula as it approaches 0
lim f(a+h) - f(a) / h h->0
lim f(x) - f(a) / x-a x->a
Dertivative
Expression that calculates the instan. RoC of a function at any given x-value
Derivative notation
Can be notated as
f’(x) “f prime of x”
y’ “y prime”
dx/dy “derivative of x with respect to y”
Official definition of derivative involving limits
Limit gives an expression that calculates instantaneous RoC of f(x) at any given point
lim f(x+h) - f(x) / h h->0
Equation of tangent line
Line tangent to curve of f(x) at x=a can be represented in point-slope form
y-y1 = m(x-x1) where y1 = y-value of the point at a x1 = x-value of point at a m = slope of the tangent line found by derivative
Differentiability
Derivative exists for each point in domain
Graph must be smooth line or curve for derivative to exist
Local linearity
Looks like a line when zoomed in
Derivative fails to exist at
Discontinuities -> jump or removable
Corner/ cusp -> sharp corners
Vertical tangent -> slope is undefined
Differentiability implies
continuity but continuity does not imply differentiability
Power rule
f(x) = x^n
f’(x) = n*x^n-1
Parallel tangent line
Tangent lines that has the same slope
To find parallel tangent lines on two different functions,
set the derivatives of the functions equal to each other and solve for 0
Derivative of a constant is
ALWAYS zero
Derivative rule: Constant multiple
d/dx cu =
c du/cx
Exp -> y = 3x^2
y’ = 3 * 2x = 6x
Derivative rule: Sum
d/dx (u + v) =
du/dx + dv/dx
Derivative rule: Difference
d/dx (u - v) =
du/dx - dv/dx
Horizontal tangent lines
Slope is always equal to zero
To determine where a function has a horizontal tangent,
Set derivative to zero and solve
Normal lines
Passes through the same point the tangent line does but it is perpendicular to tangent line
To find a normal line,
Perform steps as needed to determine a tangent line
Convert the slope of the tangent line to its negative reciprocal
To determine continuity at a point,
Set the left side and right side of the function at that point equal to each other
Substitute same point into left and right side
If sum at left is equal to sum at right, then the function is continous at that point
To determine differentability at a point
Set the derivative of the left and right sides of a function equal to each other
If evaluated to the same sum, then the function is differentiable at that point
d/dx cos(x) =
-sin(x)
d/dx sin(x) =
cos(x)
ln (1) = ln (0) = e^0 = e^lna = ln^ea=
0 undefined 1 a a
d/dx a^x =
a^xln(a)
d/dx e^x =
e^x
d/dx loga(x) =
1/x * 1/ln(a)
d/dx ln(x) =
1/x
Product rule
h(x) = f*g
h’(x) = [f’(x)g(x)] + [[f(x)g’(x)]
Quotient rule
h(x) = f/g
h’(x) = [f’(x)g(x)] - [[f(x)g’(x)] / g^2
d/dx tan(x) =
sec^2(x)
d/dx cot(x) =
-csc^2(x)
d/dx sec(x) =
sec(x)tan(x)
d/dx csc(x) =
-csc(x)cot(x)
To evaluate a derivative with a calculator,
MATH -> nDeriv(function, x, specific point)
If a point is not in the domain of f, then
it is not in the domain of f’.
