AP Calculus Study Set

Derivative of e^x

e^x

Derivative of csc(x)

-csc(x)cot(x)

Derivative of cot(x)

csc²(x)

derivative of sec(x)

sec(x)tan(x)

Derivative of tan(x)

sec²(x)

derivative sin(x)

cos(x)

derivative of cos(x)

-sin(x)

derivative of arcsin(x)

1/sqrt(1-x²)

derivative of arccos(x)

-1/sqrt(1-x²)

derivative of arctan(x)

1/(1+x²)

derivative arcsec(x)

1/(|x|sqrt(x²-1))

derivative arccsc(x)

-1/(|x|sqrt(x²-1))

derivative of arccot(x)

-1/(1+x²)

derivative of a common log (log base a of x)

1/(x*ln(a))

derivative of natural log x (ln(x))

1/x

what is the power rule, chain rule, and quotient rule

The power rule states that if you have a function a^n then the derivative is na^(n-1). The chain rule states that you take a derivative of the outer function like in sin(x^5) by substituting in a u so that it becomes sin(u). You then take the derivative (dy/dx)(du/dx) so that it becomes (5x^4)cos(x^5). The quotient rule is that long formula for division f’(x)g(x)-g’(x)f(x)/(g(x))²

What is the fundemental theorem of calculus part 1?

if a continous function f(x) on the interval [a,b] has an anti-derivative/integral, F(x), that is defined on the given interval [a,x] then the derivative of F(x) will be equal to f(x)

What is the fundemental theorem of calculus part 2?

∫_a^b f(t) dt is F(t)|_a^b = F(b) - F(a)

first and second derivative test

first derivative test = function has a local maximum or minimum at a given point if f’(x) = 0 and the sign of f’(x) flips. The second derivative test states that if the second derivative f’’(x) is positive then the given point is a local minimum otherwise it is a local maximum if f’’(x) is negative. If f’’(x)=0 then the test is inconclusive.

Intermediate value theorem IVT

if on the interval [a,b] a continuous function f(x) has a value c between the y values of f(a) and f(b) then f(x) must equal c at one point.

Mean value theorem

states that if a function is differentiable on the open interval (a,b) and continuous on the closed interval [a,b], then there must exist a point where the mean value of the function f’(a)+f’(b)/a-b is equal to f’(x) at one point. THE DERIVATIVE NOT THE ACTUAL VALUE f(x)

Extreme value theorem

If a function is continuous on a closed and bounded interval then the function will have both a maximum and a minimum on that interval.