-b±√(b²-4ac)

--------------

2a

With respect to derivatives, what is the product rule?

d/dx f(x)g(x) = ?

[f(x)’·g(x) + f(x)·g(x)']

Derivatave of the first times the second

plus

the first times the derivative of the second

With respect to derivatives, what is the quotient rule?

                             d/dx f(x)

                                    ----

                                    g(x) = ?

f(x)'·g(x) - f(x)·g(x)'

------------------------

[ g(x) ]²

Derivative of the first, times the second,

minus

the first, times the derivative of the second,

all divided by

the second, squared.

(f^-1)'(a) = ?

The derivative of f inverse at a point (a) equals what?

1

----------------

f'(f^-1(a))

one divided by

the derivative of the function OF the inverse function at the point (a).

What is the "change of base" formula?

log_ax = ?

lnx

-----

lna

d/dx lnx = ?

What is the derivative of the log of x?

1

---

x

d/dx ln(u) = ?

What is the derivative of the log of u?

1     du

--  ·  ---

u      dx

∫¹/_x dx

or

∫x^-1 dx

1

----------

x ln(a)

Kind of like the d/dx lnx = 1/x but multiply the bottom by the natural log of the base (a).

a^x ln(a)

a^x      

-------  + c

ln(a)

What are the steps in logarithmic differentiation?

1) Take ln of both sides of the equation and use the properties of logarithms to simplify.

2) Use implicity differentiation with respect to x.

3) Solve the resulting equation for y'.

∫e^kx =

Where "k" is just a constant (number).

 e^kx

----

      k

Properties of Logarithms

log_a(xy) = ?

log_a(x) + log_a(y)

Properties of Logarithms

log_a(x/y) = ?

log_a(x) _- log_a(y)

Properties of Logarithms

log_a(x)^r = ?

= r log_a(x)

Just pop the exponent out front!

Trig Identity:

Cos²(θ) + Sin²(θ) = ?

1

Cos²(θ) + Sin²(θ) = 1

1

---------- * u'

√1 - u²

∫          1

                    -----------  du = ?

           √(1 - u²)

Sin^-1(u) + c

∫          1

                    -----------  dx = ?

           √(a² - x²)

Sin^-1(x/a) + c

- 1

----------

√(1-x²)

1

---------- * u'

1 + u²

∫        1

                  ---------- dx = ?

          x² + a²

1/a * tan^-1(x/a) + C