Calculus Derivatives Overview

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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


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