Calc 1

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29 terms · Derivatives

derivative of sin(x)

= cos(x)

derivative of cos(x)

= -sin(x)

derivative of tan(x)

= sec^2(x)

derivative of sec(x)

= sec(x) tan(x)

derivative of csc(x)

= -csc(x) cot(x)

derivative of cot(x)

= -csc^2(x)

derivative of trig functions reminders

...functions that start with "c" all have negative derivatives
...tan & cot have squares
...sec & csc are themselves x tan & cot respectively

derivative of a^x

= (a^x) (ln a)

where "a" is a constant

y = ln x = log_e_x

y' = 1/(x ln e) <------ln e = 1

ex: y = log_10_4x u= 4x y=log_10_u
y' = 1/(u ln 10) x (4)
y' = 4/(4x ln 10) = 1/(x ln 10)

velocity

is the first derivative of a position function
s(t) = position function
s'(t) = velocity, v(t)

acceleration

is the second derivative of a position function
s(t) = position function
s'(t) = v(t), velocity
s''(t) = v'(t) = acceleration, a(t)

marginal cost

is the first derivative of a cost function and tell us the cost to produce one more item

rate of change of population

first derivative of a population equation/function

derivative of arccosine x
(inverse cosine)

= -1/(sqrt 1-x^2)

derivative of arcsine x
(inverse sine)

= 1/(sqrt 1-x^2)

derivative of arctangent x
(inverse tangent)

= 1/(1+x^2)

derivative of e^u

= (e^u)(u')

derivative of a^u

= (ln a)(a^u)(u')

derivative of a function
(fixed #)

f(a+h) - f(a)
f'(a) = lim h-->0 --------------------
h

derivative of a function
(number varies)

f(x+h) - f(x)
f'(x) = lim h-->0 --------------------
h

cot

cos/sin

3 requirements for a function to becontinuous at x=a

f(x) must be defined at x=a
lim x-->a f(x) must exist
f(a) = lim x-->a f(x)

point slope form

(y - y_1) = m (x - x_1)

tan

sin/cos

Parametric equations

dy/dx = y'/x'

Horizontal asymptote

y = (numerator's leading coefficient) / (denominator's leading coefficient) = 1/1 = 1
If there isn't a variable in the numerator, the ha is at 0

Vertical asymptote

set denominator = 0 and solve

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