AB Calculus Derivative RULES

Volume of a solid created by revolution around a dashed horizontal line (y=#) when the region completely lies on the dashed line.

integral from left x to right x (pi(curve - axis)^2 dx

perpendicular distance lines (radius) will be vertical)

ALL Xs!!

d/dx (sin(x))

cos x

d/dx (sin(g(x))

g(x) is ANYTHING other than x

(cos(g(x))(g'(x))

Chain Rule:

d/dx f(g(x))

f'(g(x)) * g'(x)

Derivative of the outer function, leaving the inner function alone, times the derivative of the inner function

Product Rule:

d/dx [f(x)g(x)]

f'(x)g(x) + g'(x)f(x)

derivative of the first, times the second PLUS the derivative of the second, times the first

Quotient Rule:

d/dx [f(x)/g(x)]

[f'(x)g(x)-g'(x)f(x)]/(g(x))^2

derivative of the top, times the bottom MINUS the derivative of the bottom, times the top, all over the bottom squared

d/dx (cos(g(x))

g(x) is ANYTHING other than x

(-sin(g(x))(g'(x))

d/dx (cos(x))

-sin(x)

Derivative of ANY trig function (with something other than just x or u in the parentheses

Chain Rule!

Derivative of the trig function (leave what is in the parentheses alone), times the derivative of what was in the parentheses

d/dx (e^x)

e^x

d/dx (e^(f(x))

where f(x) is anything other than x

e^f(x) times f'(x)

rewrite, times the derivative of what is in the power

d/dx (lnx)

1/x

d/dx ln(g(x))

g'(x)/g(x)

blah'/blah

derivative of what is in the parentheses/what is in the parentheses

derivative of the argument divided by the argument

Power Rule

Bring down the power, rewrite the base, decrease the power by one, TIMES THE DERIVATIVE OF THE BASE!

Mean Value Theorem

if f(x) is continuous and differentiable:

The slope of tangent line equals slope of secant line at least once in the interval (a, b)

f '(c) = [f(b) - f(a)]/(b - a)

IROC=AROC

Rolle's Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) [DON'T INCLUDE END POINTS] where f'(c)=0

Specific case of MVT (IROC=AROC=0)

Extreme Value Theorem (EVT)

If f (x) is continuous on [a, b],

then f (x) will have an absolute max and an

absolute min in the interval (including endpoints)

Candidates Test

Critical Value

Where f'(x)=0 or undefined

Absolute Extrema

Min and/or Max

these will occur at either end points or critical values

If s(t) is position of a particle, then s'(t)=

velocity of the particle

If v(t) is velocity of a particle, then v'(t)=

acceleration of the particle

Related Rates

1. Draw a picture & label with VARIABLES for anything that will be changing measurements. Put any changing values or rates in your Find, When Given:

2. Find: A rate: d blah/dt (use variable from image)

When: a specific point (not usually a rate)

Given: d blah/dt (use variable from image)

3. Write an equation (or use the given one) that will RELATE the rates). The ONLY values allowed to be plugged in before differentiation are CONSTANTS!

4. Take the derivative WITH RESPECT to TIME! All variables will end up with a d blah/dt, and you better end up with more than one rate, as these are RELATED rates!

5. Use side work to find any missing values needed for equation, and solve.

f(x) is increasing

f'(x)>0 (is positive)

(above the x-axis)

sign test if needed

f(x) is decreasing

f'(x)<0 (is negative)

(below the x-axis)

sign test if needed

f(x) has a local maximum at x=c

f'(c)=0 or undefined and changes from + to -

(crosses from above the x-axis to below)

f(x) has a local minimum at x=c

f'(c)=0 or undefined and changes from - to +

(crosses from below the x-axis to above)

Second Derivative Test:

f(x) has a minimum at x=c if

f'(c)=0 or undefined

AND

f"(c)>0 (is positive)

Second Derivative Test:

f(x) has a maximum at x=c if

f'(c)=0 or undefined

AND

f"(c)<0 (is negative)

f(x) has a point of inflection at x=c if

f"(x)=0 or undefined AND CHANGES SIGNS!

or when f'(x) and a min/max

f(x) is concave up when

f"(x)>0

OR

f'(x) is increasing

f(x) is concave down when

f"(x)<0

OR f'(x) is decreasing

The limit of f(x) as x approaches c EXISTS if

limit of f(x) as x approaches c from the left EQUALS the limit of f(x) as x approaches c from the right

Methods for trying to find if a limit exists

1. plug in

2. factor, cancel, plug in

3. conjugates, cancel, plug in

4. pluck the function if it looks like the definition of a derivative and then take the derivative

5. L'Hopital's if indeterminate form, then plug in

f(x) is continuous at x=a if

f(a)=limit of f(x) as x approaches a.

Definition of a derivative

lim h->0 [f(x+h)-f(x)/h]

Used if asking for the derivative at ANY x-value

Alternate definition of derivative

limit x->a [f(x)-f(a)]/(x-a)

Used if asking for derivative AT a given point

A function is NEVER differentiable if

the slope of f(x) at a value coming from the left does NOT equal the slope of f(x) at a value coming from the right

1. Sharp points (cusps)

2. Vertical tangent lines

3. Points of discontinuity

If f(x) is differentiable then f(x) MUST be

CONTINUOUS

but the converse may not be true

Slope of a horizontal tangent line =

0

When the derivative (slope of the tangent line) =0

Slope of a vertical tangent line is

DNE (Undefined)

When the denominator of a derivative (slope of the tangent line) =0

Intermediate Value Theorem (IVT)

If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k

Average Rate of Change of f(x) on [a,b]

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

Slope of secant line between two points, used to estimate instantaneous rate of change at a point (usually when given a table)

Equation of a tangent line to f(x) at (a,b)

y-b=m(x-a)

where (a,b) is a point on f(x) and m=derivative at the given point

f'(3) means

1. slope of the tangent line to f(x) at x=3

2. slope of the curve of the curve, f(x), at x=3

3. instantaneous rate of change of f(x) at x=3

4. dy/dx at x=3

d/dx tanx

(secx)^2

How do you approximate f(value close to a) using equation of line tangent to f(x) at (a,b)

plug the point into the equation of the line tangent and solve

How do you know if a linear approximation f(1.2) is an overestimate or an underestimate?

Use your SECOND DERIVATIVE!

If f"(x)>0, f(x) is concave up, so the tangent line lies BELOW the curve of f(x), so an underestimate

If f"(x)<0, f(x) is concave down, so the tangent line lies ABOVE the curve of f(x), so an overestimate

Integral of f'(x) dx=

f(x) + C

Integral of f'(anything other than x) dx

u-sub what is in the parentheses, and then integrate

Integral of x^(a number)

Increase the power by 1 divided be the new power

Integral of anything other than x raised to a number

U-SUB what is being raised to the power and then integrate

integral of sinx dx

-cosx + c

integral of cosx dx

sinx + c

integral of e^x dx

e^x + C

integral of e^(anything other than x)

U-SUB whatever is in the power and then integrate

Integral from a to b f'(x)dx

f(b)-f(a)

If a function is given as a graph, you calculate the integral from a to b of the function by finding

the SIGNED AREA between the function and the x-axis

If a function is given as a graph, you can approximate the integral from a to b of the function by using a

Riemann Sum (right, left, midpoint, or trapezoidal) to approximate the signed area between the given function and the x-axis

d/dx [integral from a to x of f(t)dt]

f(x)

this is the second fundamental thm of Calculus

scribble, dribble, times the derivative of what you dribbled

d/dx [integral from a to g(x) of f(t)dt]

f(g(x)),times g'(x)

this is the second fundamental thm of Calculus

scribble, dribble, times the derivative of what you dribbled!

plug g(x) into f(t) for every t and then multiply by the derivative of g(x)

G(x)=integral from a to x of f(t)dt

OR

G(x)=integral from a to g(x) of f(t)dt

G'(x)=

This is STILL the FTCII !!!

f(x)

OR

f(g(x)),times g'(x)

WRITE IT!!

A particle is moving up or right if

v(t)>0 (is positive)

A particle is moving left or down if

v(t)<0 (is negative)

Speed =

absolute value of velocity

A particle is speeding up when

v(t) and a(t) have the SAME SIGN

Either both positive or both negative

The particle is slowing down

v(t) and a(t) have the OPPOSITE SIGNS

integral from a to b of v(t) dt means

NET change in distance traveled from the time interval a to b

given v(t), to find total distance travelled

∫ abs[v(t)] over interval a to b

given v(t) and initial position t = a, find final position when t = b

F=I+C

s(b) = s(a) +∫v(t) over interval a to b dt

Average Value of a Function

1/[b-a] [integral from a to b f(x)]dx

given a(t) and initial velocit t = a, find final velocity when t = b

F=I+C

v(b) = v(a) +∫a(t) over interval a to b dt

When do you use the average value formula?

When asked for the average value or asked for the average of whatever you are given in the problem

When do you use [f(b)-f(a)]/[b-a]

When asked for

1. average rate of the derivative of what you are given in the problem

2. to approximate a derivative - find a rate and you are not given a rate and you can't use instantaneous

Area between two curves, where distance lines are vertical

integral from left x to right x of top curve - bott curve dx

ALL Xs!!

Area between two curves, where distance lines are horizontal

integral from bot y to top y of right curve - left curve dy

ALL Ys!

Volume of a solid formed on a given region whose cross sections are perpendicular to the x-axis

integral from left x to right x of the AREA of the cross section dx

(perpendicular distance lines will be vertical)

ALL Xs!

Volume of a solid formed on a given region whose cross sections are perpendicular to the y-axis

integral from bott y to top y of the AREA of the cross section dy

(perpendicular distance lines will be horizonal)

ALL Ys!

Volume of a solid created by revolution around a dashed vertical line (x=#) when the region completely lies on the dashed line.

integral from bott y to top y (pi(curve - axis)^2 dy

perpendicular distance lines (radius) will be horizontal)

ALL Ys!!

Volume of a solid created by revolution around a dashed horizontal line (y=#) when the region does NOT completely lies on the dashed line (hole will be created)

integral from left x to right x (pi(far curve - axis)^2 - (pi(close curve - axis)^2 dx

perpendicular distance lines (radius) will be vertical)

ALL Xs!!

Volume of a solid created by revolution around a dashed vertical line (x=#) when the region does NOT completely lies on the dashed line (hole will be created)

integral from bott y to top y (pi(far curve - axis)^2 - (pi(close curve - axis)^2 dy

perpendicular distance lines (radius) will be horizontal)

ALL Ys!!

derivative of inverse function

1/f'(f^(-1)(x))

d/dx [arcsin(g(x))]

1/√(1-g(x)^2), times g'(x)

d/dx [arccos(g(x))]

-1/√(1-g(x)^2), times g'(x)

d/dx [arctan(g(x))]

1/(1+g(x)^2), times g'(x)

Solving a Differential Equation

1. Separate variables

2. Integrate both sides-- right side: + C

3. Plug in initial condition to solve for C

4. Plug C back in and solve for y

If rate of change of y varies directly with y: dy/dt = ky

This solves to:

y=Ce^(kt)

integral of dx/x or du/u

ln |x| + C

or

ln |u| + C