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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

y¹

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