Different Types of Differentiation and how to solve them

What is the derivative of log<a>x

1 / (ln a) x

What does the first derivative test solve

Relative extrema (min = negative to positive, max = positive to negative)

Derivative rule of Ln

d/dx Ln x = 1/x or d/dx ln u = u'/u

3 ways to solve integration trig properties

1) get rule to show up with u substitution 2) Use derivatives rules from 10/8 from the first lesson to get simpler u substitution 3) rewrite as the sum of the quotients by separating the numerator

Compounded n times per year equation (P = amount of deposit, t = # of years, A = balance after t years, r = annual rate interest, n = # of compounding per year)

A = P(1 + (r/n))^nt

Compounded Continuously (P = amount of deposit, t = # of years, A = balance after t years, r = annual rate interest, n = # of compounding per year)

A = Pe^rt

What does second derivative test solve

Concavity (If its positive then its concave up, if its negative then its concave down)

inverse trig on unit circle

Cos = top half of the circle, sin = right half of circle, Tan = right half of the circle, Sec = top half of circle with hole in the middle, Csc = right half with hold in the right point, cot = top half with holes on the right and left point

Implicit Differentiation steps

Differentiation both sides of the equation and put dy/dx next to any with y next to them, get all dy/dx equations to one side, get dy/dx alone

How to find derivative of equation with the x and y on same side

Do the derivative of each y value and multiply whats remaining by dy/dx. If x and y are multiplied together do product rule and have one side of the product rule have dy/dx

How to do common denominator for 3 variables

Factor, find LDC of coefficients, every variable seen with highest exponent

Definite Integrals steps

Find out the antiderivative, plug in the upper limit and subtract the lower limit from the upper limit.

Related Rates steps

Identify all variables and rates of change, sketch a diagram, write an equation to connects the variables together, differentiate both sides in terms of time

How to find the derivative when using logarithmic differentiation but the side has been multiplied by log<a>

Ln both sides, the side with the log<a> each ln will be ln a instead of just ln and will be put into the denominator of each derivative

Quotient rule for logarithmic function

Log b (M/N) = Log b M - Log b N

Product rule for logarithmic function

Log b (MN) = Log b M + Log b N

Power rule for logarithmic functions

Log b (M^P) = P x Log b M

How to find slope of tangent line

Plug point into the first derivative

How do you prove an inverse equation

Plug the inverse equation into the original equation in place of x then simplify it

How to find equation of tangent line

Plug the points into equation y - y1 = m(x-x1)

Limit Process Steps

Set Delta x to 0 and write the function f(x + delta x) - f(x) divided by delta x, then plug x - delta x into the f(x) so each x is (x - delta x) then simplify the equation down until there is no undefined answer when plugging in 0 to delta x and that's your answer

How to do inverse functions

Switch domain and range, reflect over line y = x, and notate it as f^-1 (x), make sure it passes horizontal line test

Logarithmic Differentiation

Take ln of both sides, use product rule, quotient rule, or power rule to expand it, then take derivative of both sides by using u'/u

What needs to be done to find the derivative of an expression if x is in the variable outside of ln

You need to use product rule you cant just multiply the outside of ln inside when there an x in the equation like you can with a constant

How to solve for the slope of the inverse function when given the original function and the derivative of the original function

[f^-1] ' (a) = 1/(f ' [f^-1 (a)])

Derivatives of exponential functions

d/dx (e^x) = e^x * (x') or e^x * 1 = e^x OR d/dx (a^u) = a^u) * u' * ln a

How to find derivative of a logarithmic function without base e

d/dx [a^x] , a = e^ln a . A is a constant, substitute e^ln a for a in the problem and solve the derivative then substitute a back into the answer

Antiderivatives of exponential functions

e^u dx = ((e^u)/u ' ) + C or e^2x dx = e^2x/2 + C

How to make a function a natural logarithmic function

find u, find the derivative of u, get dx alone, put du out in front, find the antiderivative of the integral, keep du out in front and substitute u

How to do chain rule for exponential functions

find v(x), v ' (x), u (x) and u ' (x) and find the derivative using v ' (u(x)) * u ' (x). Plug u(x) back into the final answer

Log properties

log<a>x = ln x / ln a log<a>b = log<c>b / log<c>a (calculator would recommend c = 10) log<e>x = ln x log<10>100 = 2

How do you solve x when e^x = 12

make x ln 12 so e^ln 12 = 12

How to make an inverse equation

replace x with y and y with x in the original equation then get y alone

Arccos u property

u = - u ' / (square root)1 - u^2

Arccot u property

u = - u ' / 1 + u^2

Arcsin u property

u = u ' / (square root)1 - u^2

Arctan u property

u = u ' / 1 + u^2

Integral arctan property

{du / a^2 + u^2 = 1/a arctan(u/a) + C

Integral acrsin property

{du / a^2 - u^2 = arsin(u/a) + C