Chapter 2: Derivatives

d/dx cotx

-csc^2(x)

d/dx cscx

-cscxcotx

d/dx cos

-sinx

Implicit Differentiation

A method for taking the derivative when the function is NOT set = to y, and it would be too difficult to solve for it.

Quotient Rule

A method for taking the derivatives when one function is divided by another. f'(x)·g(x) + g'(x)·f(x)

Product Rule

A method for taking the derivatives when one function is multiplied by another. [f'(x)·g(X)-g'(x)·f(x)]/[g(x)^2]

Chain Rule

A method for taking the derivatives where one takes the derivative of the outside, leaving in the inside alone, times the derivative of the inside. This method is necessary for composition of functions, where something exists "inside" another. f'(g(x))·g'(x)

Power Rule

A method for taking the derivatives where the coefficient is multiplied by the power, and the power is reduced by one.

Secant Line

Average velocity/rate of change; gives the slope between two points on a curve

Tangent Line

Instantaneous velocity/rate of change; gives the slope at exactly one point on a curve

Derivative

Instantaneous velocity/rate of change; the slope of a curve at exactly one point; the slope of the tangent line.

Twist

One of the three fails for a derivative; an undefined slope

Sharp point

One of the three fails for a derivative; the slope from the left doesn't approach the slope from the right of the limit; the limit from the left of f'(x) doesn't approach the limit from the right of f'(x)

d/dx sinx

cosx

Velocity at Impact

find by setting the position equation = o, solving for time, and plugging that back into the velocity equation for t.

Maximum Height

find by setting the velocity equation = o, solving for time, and plugging that back into the position equation for t.

d/dx tanx

sec^2(x)

d/dx secx

secxtanx

Average Velocity

slope of a secant line; ∆s/∆t

Instantaneous Velocity

slope of a tangent line; the derivative

Definition of Derivative (one method)

the limit as x approaches 0 of [f(x+△x)-f(x)]/[△x]

Alternative Definition of Derivative of a Point

the limit as x approaches c of [f(x)-f(c)]/[x-c]