Studied by 19 people
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)
Pretend inner function is not there and work on the outer function. Find its derivative
Then find derivative of inner function
Expand/simplify if needed
Think levels.
Finding the Equation of a Tangent Line to a Curve
Determine f(x)
Determine f’(x)
Determine f’(x) at given point
Write equation in point-slope form (y - y1 = m(x - x1)
Finding Derivative at x=c
Determine c and type of derivative form (standard, modified, or alternative?)
Determine function
Determine derivative of function
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
Find intervals where function is increasing and/or decreasing
Determine slope (is m > or < 0?)
Determine horizontal tangent line
Determine position of f’(x) (will the slope be above or below the x-axis?)
Graph derivative based on identifications
Finding points where there’s a horizontal tangent line
Find f’(x)
Determine zero(es)
Plug x-value(s) into function f(x)
Write the point(s)
Finding points where there’s a vertical tangent line
*Remember that vertical tangent lines are when the slope is undefined
Find f’(x)
Find x in terms of y (if derivative is a fraction, make denominator = 0 and find x)
Plug x into original function and solve for y
Plug y into original equation and solve for x
Write point
Finding derivatives of trigonometric functions
If needed/possible
Simplify expression
Express everything w/ cosine and sine
Factor
Rewrite after applying trigonometry identities
Piecewise functions and finding values for a and b
Continuity Test (equate)
Use differentiability to find a
Go back to the equated expression and plug in a, solve for b
Differentiability Test
Differentiate the two functions given in piecewise function and equate them
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
Plug x into original function to find value(s) for y
Use implicit differentiation on the function to find the slope(s)
Write the equation of tangent/normal line(s)
Derivative of Inverse Functions
Find the point, x-value
Find the derivative of the function
Plug x-value into derivative
Find the derivative of the inverse function using reciprocal
Derivative of Inverse Functions Formula
g’(x) = 1/f’(g(x))
