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The Derivative at a given point
Is equal to the change in y over the change in x
Definition of a Derivative at a Point
f¹(c)= lim x→c of f(x)- f(c)/ x- c
Slope of a tangent Line
The derivative of a function at a point equals the slope of the tangent line to the graph of the function at that point. Both equal the instantaneous rate of change.
f(c+h)−f(c)÷(c+h)−c
Forward Difference Quotient
f(c-h)−f(c)÷(c-h)−c
Backward Difference Quotient
Symmetric Difference Quotient
i.e. [f(x+h) - f(x-h)] / 2h
Definition of a Derivative as a Function
f¹(c)= lim h→0 of f(x+h)- f(x)/ h
Property: Derivative of the Power Function
If f(x)=Xⁿ, then f¹(x)= nxⁿ⁻¹, the exponent n is a constant
Differentiating the Power Function
To differentiate the power function, f(x)= xⁿ, multiply by the original exponent, n, then reduce the exponent by 1 to get the new exponent.
Derivative of a Sum of Two Functions
The derivative of a sum equals the sum of the derivatives
Derivative of a Constant times a Function
The derivative of a constant times a function equals the constant times the derivative of the function
Derivative of a Constant Function
Constants don't change, so their rate of change is zero
Is read "y prime" (a short form of f¹(x)
dy/dx
Is read :dy, dx" (a single symbol, not a fraction)
d/dx(y)
Is read "d, dx, of y" (an operation done on y)
Definition: Antiderivative, or Indefinite Integral
Function g is an Antiderivative (or indefinite integral0 of function f if and only if g¹(x)= f¹(x)
velocity
The rate at which distance is changing with time.
acceleration
a=v¹= dx/dt
Speed
D / T
d/dx(sin x)
cos x
d/dx(cos x
-sin x
Chain Rule
y = cos²(3x)
Limit of (sin x)/x
lim x→0 of sin x/ x = 1
y=C+A cos B(x-D)
General equation of a sinusoidal function
C is the vertical translation
The sinusoidal axis is the line y=C
The amplitude equals |A|
A is the vertical dilation factor
The period equals 2π×1/|B|
1/B is the horizontal dilation factor
The phase displacement equals D
D is the horizontal translation
f¹(x)= e^x
The Derivative of f(x)=e^x
The log of a power
Equals the exponent times the log of the base
The log of a product
is the sum of the logs
The log of a quotient
is the difference of the logs
f¹(x)=1/x
The Derivative of f(x)=lnx
Then f⁻¹(x)=e^x
If f(x)=ln x
If f(x)=e^x
then f⁻¹(x)=lnx
lne^x=x
e^lnx=x
If f(x)=b^x
Then f¹(x)=b^x lnb