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