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Level 579

Derivatives, Antiderivatives & Indefinite Integral

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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.
Forward Difference Quotient
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)
Is read :dy, dx" (a single symbol, not a fraction)
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)
The rate at which distance is changing with time.
a=v¹= dx/dt
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
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
If f(x)=b^x
Then f¹(x)=b^x lnb