∫dx
∫uⁿ du
∫e^u du
∫b^u*ln(b) du
∫b^u du
∫cosu du
∫-sinu du
∫sinu du
∫sec²u du
∫-cscu*cotu du
∫cscu*cotu du
∫secu*tanu du
∫-csc²u du
∫csc²u du
∫du/√(1-u²)
∫-du/√(1-u²)
∫du/(1+u²)
∫-du/u√(1-u²)
∫du/u√(1-u²)
∫-du/(1+u²)
∫du/u
∫tan u du
-ln|cos u|+C
∫cot u du
ln|sin u|+C
∫sec u du
ln|sec u+tan u|+C
∫csc u du
-ln|csc u+ cot u|+C
f(x)g'(x) + g(x)f'(x)
derivative of the multiplaction of two functions
d/dx [f(x)/g(x)] =
quotient rule
f'(g(x))g'(x)
chain rule
d/dx [ln|x|] =
d/dx [a^x] =
d/dx [log_a X] =
∫- sinx
∫secxtanx
∫- cscxcotx
d/dx [tanx]
d/dx [cotx]
∫cosx
quotient rule
derivative of a power
chain rule
derivative of an exponential function
derivative of a natural log
d/dx [a^x]
derivative of a log base a
d/dx [sinx]
d/dx [cscx]
d/dx [cosx]
d/dx [secx]
d/dx [tanx]
d/dx [cotx]
d/dx [arcsinx]
d/dx [arccscx]
d/dx [arccosx]
d/dx [arcsecx]
d/dx [arctanx]
d/dx [arccotx]
product rule
f is continuous at x=c if...
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
Global Definition of a Derivative
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
∫-sin(x)
∫sec²(x)
∫-csc²(x)
∫sec(x)tan(x)
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
Rolle's Theorem
Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
Mean Value Theorem
The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.
First Derivative Test for local extrema
Point of inflection at x=k
Combo Test for local extrema
Horizontal Asymptote
L'Hopital's Rule
x+c
sin(x)+C
-cos(x)+C
tan(x)+C
-cot(x)+C
sec(x)+C
-csc(x)+C
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
Fundamental Theorem of Calculus #2
Mean Value Theorem for integrals or the average value of a functions
ln(x)+C
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
ln(sinx)+C = -ln(cscx)+C
ln(secx+tanx)+C = -ln(secx-tanx)+C
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
If f and g are inverses of each other, g'(x)
Exponential growth (use N= )
Formula for Disk Method
Axis of rotation is a boundary of the region.
Formula for Washer Method
Axis of rotation is not a boundary of the region.
Inverse Secant Antiderivative
Inverse Sine Antiderivative
Derivative of eⁿ
ln(a)*aⁿ+C
Derivative of ln(u)
Antiderivative of f(x) from [a,b]