Derivatives
Definition
The derivative of a function f(x) at a certain point x, is the slope of the tangent line to that function f(x) at the indicated point x.
Definition
Formula f'(x) = lim h-> 0 ( f(x+h) - f(x) ) / h
Reminders
Solving
1st. Replace your variables with the formula.
2nd. Multiply and simplify.
3rd. Replace the h(s) with 0.
Secant line: is a straight line joining two points on a function.
Tangent line: A line that touches a curve at a point without crossing over.
(a + b) = a^2 + 2ab + b^2
(a + b)^3 = a^3 + 3ab^2 + 3a^2b + b^3
y = mx + b Where m is the slope.
DON'T FORGET that the x will be replaced with f(x + h) AND f(x). So, it will be like replacing the x twice but with a different part of the formula.
Rules of Differentiation
If y=f(x) + g(x) then y'= f'(x) + g'(x)
if y=c is constant y'= 0
If y= k*f(x) then y'= f'(x)
Power Rule If y= x^n then y'= nx^(n1)
NOT Calculus Reminders
1 / x^n = x^-n
^n(square root)x^m = x^(m/n)
x^n * x^n = x^(n+n)
x^m / x^n = x^(m-n)
a(b+c) = ab + ac
Finding Equation by Tangent Line
1st. Find f'(x)
2nd. Find slope m= f`(x)
3rd. Find coordinate by substituting in original function
4th. Find equation with y=mx + b
Chain Rule
y= x^n = variable^number
y= U^n then y'= nU^(n-1) 'U U is a function in terms of x*
Differentiability
3 cases in which a function is non differentiable at certain point
1) Discontinuous
2) Function has a vertical tangent line.
Mostly ^3 (square root) x, ^5 (square root) x, ^7 (square root) x
3) Where there's a "sharp turn" or "sudden break". Mostly absolute value.
Product Rule
If a function is of the type y= f(x) * g(x), where both f(x) and g(x) are functions of x we use product rule to differentiate.
Recommend to arrange: f(x) ---- f'(x)
g(x) ---- g'(x)
y'= g(x)f'(x) + g'(x)f(x)
Quotient Rule
Where both f(x) and g(x) are functions of x
Recommend to arrange: f(x) ---- f'(x)
g(x) ---- g'(x)
y'= (g(x)f'(x) / g'(x)f(x)) / (g(x)^2)
Position - Velocity - Acceleration
Postion -> x(t) = p(t) = h(t) Velocity -> v(t) = p'(t) Acceleration -> a(t) = v'(t)
Exponential Functions
Instantaneous rate change.
y = 2^x = base^variable Mostly base y = e^x
If y= e'U then y'= U'*e^U
Particular Case y=e^x then y'= e^x
Arbitrary Base y= a^U then y'= U'(lna)a^U
Logarithmic Functions
Generally
y= lnU then y'= U' / U
Particular Case
y= lnx then y'= 1/x
Properties
lna^n = lna
ln(a*b) = lna + lnb
ln (a/b) = lna -lnb
y= logaU then y'= U' / ((lna)U)
Trigonometric Functions
f(x)= e^U -> f'(x)= U'e^U
f(x)= sin(U) -> f(x)= U'cos(U)
f(x)= cos(U) -> f'(x)= -U'sin(U)
f(x)= tan(U) -> f'(x)= U'sec^2(U)
f(x)= cot(U) -> f'(x)= -U'csc^2(U)
f(x)= sec(U) -> f'(x)= U'sec(U)tan(U)
f(x)= csc(U) -> f'(x)= -U'csc(U)cot(U)
Identities
cscU= 1 / sinU
cotU= 1 / tanU
tanU= sinU / cosU
secU= 1 / cosU
cotU= cosU / sinU
sinU + cosU = 1
Higher Order
Original function y= f(x)
1st. Derivative y'= f'(x) = dy / dx = (d(f(x))) / dx
2nd Derivative (the rate in which the rate is changing) y''= f''(x) = d^2y / d^2x = (d^2(f(x))) / dx^2
3rd. Derivative y'''= f'''(x) = d^3y / dx^3 = (d^3(f(x))) / dx^3 nthDerivative y^(n) = f(n)(x) = dy^n / d^nx = (d^n(f(x))) / dx^n