46 terms

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