AP Calculus AB - Chapter 2

Differentiation

Average velocity

• change in position/change in time

• delta s/delta t

• [s(t2) - s(t1)]/t2 - t1

Average rate of change

• change in y/change in x

• delta y/delta x

• [y(x2) - y(x1)]/x2 - x1

Difference Quotient

[f(x) - f(c)]/x -c

Secant Lines and Average Rate of Change

Average Rate of Change (passes through two points)

[f(b) - f(a)]/b - a

Tangent Lines and Instantaneous Rate of Change

(passes through one point)

[f(a) - f(a)]/a - a = 0/0

The limit of [f(x+h) - f(x)]/h as h→0 is called the derivative of f(x) (f’(x), dy/dx, y’, y’(x), df(x)/dx, d(x^n)/dx), the slope of the tangent line, and/or the instantaneous rate of change.

Limit Definition of Derivative

f’(x) = lim of [f(x+h) - f(x)]/h as h→0 = a function

Modified Form of the Limit Definition

f’(c) = lim of [f(c+h) - f(c)]/h as h→0 = a number

Alternative Form of the Limit Definition

f’(c) = lim of [f(x) - f(c)]/x - c = a number

Finding Derivatives Using the Power Rule

d(x^n)/dx = nx^n-1

Finding Derivatives Using Product Rule

(d/dx)(uv) = (u)(derivative of v) + (v)(derivative of u)

Finding Derivatives Using Quotient Rule

(d/dx)(u/v) = [(v)(derivative of u) - (u)(derivative of v)]/(v)²

Finding Derivatives Using Chain Rule

*Based on composite functions

(d/dx)(f(g(x))) = f’(g(x)) * g’(x)

1. Pretend inner function is not there and work on the outer function. Find its derivative

2. Then find derivative of inner function

3. Expand/simplify if needed

Think levels.

Finding the Equation of a Tangent Line to a Curve

1. Determine f(x)

2. Determine f’(x)

3. Determine f’(x) at given point

4. Write equation in point-slope form (y - y1 = m(x - x1)

Finding Derivative at x=c

1. Determine c and type of derivative form (standard, modified, or alternative?)

2. Determine function

3. Determine derivative of function

4. Find derivative at x=c

Normal Lines

To find the normal line, rewrite the tangent line except the slope (m) becomes the negative reciprocal.

Differentiability and Continuity

Derivatives of one-sided limits must exist and be equal to each other, differentiable on closed interval [a,b].

A function is not differentiable at a point when the graph has:

• Sharp turn/cusp

• Vertical tangent line

• Discontinuity

Graphing Derivatives

1. Find intervals where function is increasing and/or decreasing

2. Determine slope (is m > or < 0?)

3. Determine horizontal tangent line

4. Determine position of f’(x) (will the slope be above or below the x-axis?)

5. Graph derivative based on identifications

Finding points where there’s a horizontal tangent line

1. Find f’(x)

2. Determine zero(es)

3. Plug x-value(s) into function f(x)

4. Write the point(s)

Finding points where there’s a vertical tangent line

*Remember that vertical tangent lines are when the slope is undefined

1. Find f’(x)

2. Find x in terms of y (if derivative is a fraction, make denominator = 0 and find x)

3. Plug x into original function and solve for y

4. Plug y into original equation and solve for x

5. Write point

Finding derivatives of trigonometric functions

If needed/possible

1. Simplify expression

2. Express everything w/ cosine and sine

3. Factor

4. Rewrite after applying trigonometry identities

Piecewise functions and finding values for a and b

1. Continuity Test (equate)

2. Use differentiability to find a

3. Go back to the equated expression and plug in a, solve for b

Differentiability Test

1. Differentiate the two functions given in piecewise function and equate them

2. Plug in x if given and see if the values are equal to each other. If not, then it’s not differentiable.

Implicit Differentiation

When the variables aren’t the same, continue to apply any rules when applicable (Power, Product, Quotient, Chain) and make it explicit. (dy/dx)

Implicit Differentiation and finding tangent/normal lines

1. Plug x into original function to find value(s) for y

2. Use implicit differentiation on the function to find the slope(s)

3. Write the equation of tangent/normal line(s)

Derivative of Inverse Functions

1. Find the point, x-value

2. Find the derivative of the function

3. Plug x-value into derivative

4. Find the derivative of the inverse function using reciprocal

Derivative of Inverse Functions Formula

g’(x) = 1/f’(g(x))