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Calculus Unit 2 Derivatives Overview
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Formula for the Derivative by the Limit Process
f'(x)=lim ∆x→0[(f(x+∆x)-f(x))/(∆x)]
The formula for the deriative by the limit process comes from...
The tangent line problem
Point-Slope Formula
y - y₁=m(x - x₁)
If an equation is NOT continuous, then it is ____ _________.
Not Differentiable
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
An equation must be ______________ to be differentiable, but being _____________ does not guarantee differentiablility.
Continuous, Continuous
A derivative is a graph of the ___________ of the _________ line at each point.
Slope, Tangent
d/dx means taking the derivative with _________ to ____.
Respect, X
The derivative of a constant is always = _____
0
Sum Rule: d/dx[f(x) + g(x)]=
f'(x) + g'(x)
Difference Rule: d/dx[f(x) - g(x)]=
f'(x) - g'(x)
Power Rule: d/dx[ xⁿ]=
nx^(n-1)
You can only use the power rule when the variable is in the _____________
Numerator
You can only use the power rule when there are NOT ________ NOR _________ of variables
Products, Quotients
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
The derivative of position is
Velocity
The derivative of velocity is
Acceleration
s(t) usually represents
Position
s'(t) usually represents
Velocity
s''(t) usually represents
Acceleration
v₀ represents
Initial velocity (starting velocity)
s₀
Initial height (starting height)
d/dx[sin(x)]=
cos(x)
d/dx[cos(x)]=
-sin(x)
d/dx[tan(x)]=
sec²(x)
d/dx[sec(x)]=
sec(x)tan(x)
d/dx[csc(x)]=
-csc(x)cot(x)
d/dx[cot(x)]=
-csc²(x)
Product Rule (Mnemonic)
1D2+2D1
d/dx[f(x)g(x)]=
f(x)g'(x)+g(x)f'(x)
Quotient Rule (Mnemonic)
LowDHi-HiDLow over the square of what's below
d/dx[f(x)/g(x)]=
[g(x)f'(x)-f(x)g'(x)]/g(x)²
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)
Chain Rule: d/dx[f(u)]=
f'(u)u'
Chain Rule: d/dx[f(g(x))]=
f'(g(x))g'(x)
Never leave answers with _________ exponents
Negative
Never leave answers with _________ fractions
Complex
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
Explicit Form means that the equation is in terms of _____
y=
Implicit Form means that there are x's and y's on the _______ _______ of an equation.
Same Side
As you take a derivative of y with respect to x, you have to multiply the derivative by ________
dy/dx
In implicit differentiation, we multiply by dy/dx because of the _________ rule
Chain
d²y/dx² means to find the __________ ____________
Second Derivative
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
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
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