Derivative rules

y'all you'd better memorize all of this. it's literally everything calculus is made up of. if you don't know any of these... well... good luck charlie!

d/dx (c) = 0

derivative of a constant

d/dx (xⁿ) = nxⁿ⁻¹

Power rule (derivative of a power)

d/dx (f(x)g(x)) = f(x)g’(x)+g(x)f’(x)

product rule

d/dx (f(x)/g(x)) = (g(x)f’(x)-f(x)g’(x))/(g(x))²

quotient rule

d/dx (f(g(x)) = g’(x)(f’(g(x))

chain rule

d/dx (sinx) = cosx

derivative of sine

d/dx (cosx) = -sinx

derivative of cosine

d/dx (tanx) = sec²x

derivative of tan

d/dx (cotx) = -csc²x

derivative of cotangent

d/dx (secx) = tanxsecx

derivative of secant

d/dx (cscx) = -cscxcotx

derivative of cosecant

dy/dx f⁻¹(x) = 1/(f⁻¹(x))

derivative of an inverse

d/dx eⁿ = eⁿ

derivative of eⁿ (n meaning the whole exponent)

d/dx aⁿ = aⁿ(ln(a))n’

derivative of a number at a variable exponent (n meaning the whole exponent)

d/dx (ln(x)) = 1/x

derivative of ln(x)

d/dx (ln(u))= u’/u

derivative of a natural log (the whole exponent)

d/dx (logₙu) = (u’/uln(n))

derivative of the log of a number

d/dx (sin⁻¹x) = 1/(√(1-x²) )

derivative of arcsine

d/dx (cos⁻¹x) = -1/(√(1-x²))

derivative of arccosine

d/dx (tan⁻¹x) = 1/(1+x²)

derivative of arctan

d/dx (csc⁻¹x) = 1/(x√(x²-1))

derivative of arccosecant

d/dx (sec⁻¹x) = -1/(x√(x²-1))

derivative of arcsecant

d/dx (cot⁻¹x) = -1/(1+x²)d

derivative of arccotan

f’(x) = lim x→0 ((f(x+h)-f(x))/h)

definition of a derivative

you find the derivative of the function and plug in that point. (if you are only given the x value, plug it into the original function).

how do you find slope of the tangent line to a function at a point?

(f(b)-f(a))/(b-a) or ∆y/∆x

average rate of change (aka the way to find the derivative of the function at a secant line)

if f is continuous on \[a,b\] and d is any number between f(a) and f(b), then there is at least one number c between a and b such that f(c) = d

How do you use the intermediate value theorem?

if f(x) is continuous and if \[a,b\] and f(x) is differentiable (a,b), then there exists at least one c on (a,b) such that f’(c) = (f(b)-f(a))/(b-a)

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1) find f(a) and f(b) and plug it into the expression

2) find the derivative of the given function. set the derivative equal to f’(c) and find c

how do you use the mean value theorem?

if a function is continuous on a closed interval, then the function is guaranteed to have and absolute maximum and absolute minimum in the interval

how do you use the extreme value theorem?

take the slope of the tangent line and the point and plug it into point slope formula (y-y₁ = m(x-x₁))

how do you find the equation of a tangent line?

take the slope of the tangent line and find the line perpendicular to the slope (negative reciprocal). then put it into the point-slope formula and use the same points

how do you find the equation of a normal line?

s(t)

position of a particle

s’(t)

velocity of a particle (aka v(t))

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finds how quickly something is moving and it’s direction

s”(t)

acceleration of a particle (aka a(t) or v’(t))

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finds the rate of change of the velocity

if velocity < 0

when does the particle move left?

if velocity > 0

when does the particle move right?

speed is increasing

if velocity and acceleration have the same sign

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you can find this by using the number line method (stack two number lines on top of each other, one for velocity, one for acceleration) then seeing when the lines have the same signs at a certain point

speed is decreasing

if velocity and acceleration have different signs

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you can find this by using the number line method (stack two number lines on top of each other, one for velocity, one for acceleration) then seeing when the lines have the same signs at a certain point

velocity = 0

acceleration ≠ 0

the particle has stopped moving and is changing direction

|v|

speed

(pertaining to velocity and acceleration)

the slope of the line segment connecting the points for the time interval \[a,b\] (plug a and b into x(t) to find their respective y values and use the slope formula)

average velocity (you are NOT plugging this into x’(t) because by using the slope formula, you are already finding the derivative of x(t))

(pertaining to velocity and acceleration)

the slope of the line segment connecting the points for the time interval \[a,b\] (plug a and b into x’(t) to find their respective y values and use the slope formula)

average acceleration (you are NOt plugging this into x”(t) because by using the slope formula, you are already finding the derivative of x’(t))

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how much the particle is changing direction from a to b

(pos: moving right, neg: moving left)

the total distance traveled from a to b

(you can do this by hand by finding the places where the particle stops then doing as shown in the photo (key: c and d are the sample stopping points)

1) find the derivative of the given function and set it equal to zero

2) solve for the variable

3) find the second derivative to see if it’s a max/min

how do you solve for a word problem asking for the max/min

(don’t forget to write “local extrema occur at critical numbers which are when f’=0 or f’ DNE“ or “absolute extrema occur at endpoints or critical numbers which are when f’=0 or f’ DNE’)

When f’(c)=0 or f’ is undefined at c

critical numbers

occur at critical numbers ONLY

relative/local extrema

occur at critical numbers AND endpoints

absolute extrema

first derivative test

1) relative minimum

2) relative maximum

let c be a critical number of a function f that is continuous on an open interval i containing c. if f is differentiable on the interval, except possibly at c, then f(c) can be classified as:

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1) if f’(x) changes from negative to positive at c (or f changes from decreasing to increasing, then f(c) is a __________________ of f.

2) if f’(x) changes from positive to negative at c (or f changes from increasing to decreasing, then f(c) is a __________________ of f.

second derivative test

1) relative minimum

2) relative maximum

Left f be a function such that the second derivative of f exists on an open interval containing c. (Testing critical numbers in the second derivative)

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1) if f’(c) = 0 and f”(c) > 0, then f(c) is a _____________.

2) if f’(c) = 0 and f”(c) < 0, then f(c) is a _____________.

test for concavity

1) upward

2) downward

Let f be a function for who second derivative exists on an open interval i. Find the x values where f“(x) = 0 or f”(x) is undefined. Evaluate values to the left and right.

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1) if f”(x) > 0 for all x in i, then the graph of f is concave _______ in i.

2) if f”(x) < 0 for all x in i, then the graph of f is concave _______ in i.

definition of an inflection point

A function has an inflection point at (c, f(c))…

1) if f”(c) = 0 or f”(c) DNE

2) if **f”(x)** changes sign from positive to negative or negative to positive at x = c

***OR***

if **f’(x)** changes from increasing to decreasing or decreasing to increasing at x = c (aka has a hill/valley)

***OR***

if **f(x)** changes concavity at x = c

(f⁻¹(x))’ = 1/(f’(f⁻¹(x)) , f’(f⁻¹(x)) ≠ 0

***OR***

if g(x) is the inverse of f(x): g’(x) = 1/(f’(g(x))) , f’(g(x)) ≠ 0

derivative of an inverse function

If lim x→c f(x) = 0 and lim x→c g(x) = 0, then by L’Hopital’s rule…

lim x→c f(x)/g(x) = lim x→c f’(x)/g’(x) = lim x→c f’(c)/g’(c)

L’hopital’s Rule