Calculus Derivatives Overview
The formula for the deriative by the limit process comes from...
The tangent line problem
The derivative of position is
Velocity
s'(t) usually represents
Velocity
d/dx[f(x)/g(x)]=
[g(x)f'(x)-f(x)g'(x)]/g(x)²
d/dx[sin(x)]=
cos(x)
As you take a derivative of y with respect to x, you have to multiply the derivative by ________
dy/dx
Chain Rule: d/dx[f(g(x))]=
f'(g(x))g'(x)
Chain Rule: d/dx[f(u)]=
f'(u)u'
Sum Rule: d/dx[f(x) + g(x)]=
f'(x) + g'(x)
Difference Rule: d/dx[f(x) - g(x)]=
f'(x) - g'(x)
Formula for the Derivative by the Limit Process
f'(x)=lim ∆x→0[(f(x+∆x)-f(x))/(∆x)]
d/dx[f(x)g(x)]=
f(x)g'(x)+g(x)f'(x)
f'''(x) means the third derivative. We use tick marks to represent the level of all derivatives up to the third derivative. But how do we write the fourth derivative?
f(⁴)(x)
Power Rule: d/dx[ xⁿ]=
nx^(n-1)
d/dx[sec(x)]=
sec(x)tan(x)
d/dx[tan(x)]=
sec²(x)
If a graph of an equation has a horizontal tangent line (slope of zero), then the graph of its derivative should cross the ___-_______ at that point
x-axis
Point-Slope Formula
y - y₁=m(x - x₁)
Explicit Form means that the equation is in terms of _____
y=
d/dx[csc(x)]=
-csc(x)cot(x)
d/dx[cot(x)]=
-csc²(x)
d/dx[cos(x)]=
-sin(x)
The derivative of a constant is always = _____
0
Product Rule (Mnemonic)
1D2+2D1
The derivative of velocity is
Acceleration
s''(t) usually represents
Acceleration
In implicit differentiation, we multiply by dy/dx because of the _________ rule
Chain
Never leave answers with _________ fractions
Complex
An equation must be ______________ to be differentiable, but being _____________ does not guarantee differentiablility.
Continuous, Continuous
The chain rule states that if you take the derivative of the outside, and the inside is differentiable, then you have to multiply by the ________ of the __________
Derivative, Inside
You can only use the power rule when there are NOT _________ outside a parenthesis that would cause a variable to have a higher degree
Exponents
s₀
Initial height (starting height)
v₀ represents
Initial velocity (starting velocity)
Quotient Rule (Mnemonic)
LowDHi-HiDLow over the square of what's below
Never leave answers with _________ exponents
Negative
If an equation has a sharp turn, then it is _____ ____________.
Not Differentiable
If an equation has a vertical tangent line, then it is ______ ______________
Not Differentiable
If an equation is NOT continuous, then it is ____ _________.
Not Differentiable
You can only use the power rule when the variable is in the _____________
Numerator
s(t) usually represents
Position
Most of the problems in chapter 2 will use a combination of the power rule, along with these 3 rules.
Product Rule, Quotient Rule, Chain Rule
You can only use the power rule when there are NOT ________ NOR _________ of variables
Products, Quotients
d/dx means taking the derivative with _________ to ____.
Respect, X
Implicit Form means that there are x's and y's on the _______ _______ of an equation.
Same Side
d²y/dx² means to find the __________ ____________
Second Derivative
A derivative is a graph of the ___________ of the _________ line at each point.
Slope, Tangent
