Calculus One Differentiation and Derivations and its Appliances
derivative for arccos x
-1/square root of (1-x^2)
derivative for cotx
-csc^2x
derivative for csch x
-csch x coth x
derivative for coth x
-csch^2 x
derivative for csc x
-cscx cotx
derivative for sech x
-sech x tanh x
derivative for cosx
-sinx
derivative for arctan x
1/(1+x^2)
derivative for arctanh x
1/(1-x^2)
derivative for arcsinx
1/square root of (1-x^2)
derivative for arcsinh x
1/square root of (x^2+1)
derivative for arccosh x
1/square root of (x^2-1)
derivative for ln(x)
1/x
How to solve critical numbers of function f(x)=^3 -3x +5
Find the derivative f'(x)=3x^2 -3 3x^2-3=0 x= 1 or -1 and they both are critical numbers
How to find the derivative of y=x^x when x is greater than zero
If y=x^x and x is greater than zero then y=ln(x^x) ln y=x ln x use chain rule y'(1/y)=ln x+x(1/x)=lnx+1 where y'=dy/dx y'=(lnx +1)y times by y y'=(ln x +1)x^x there was a substitution of y to x^x
What is the concavity of a quadratic function
It tells which direction usually up or down the graph curves
How to solve a inverse derivative for f(x)=x arcsin x
Use the product rule of sums and f'(x)=(1/square root of (1-x^2)+ arcsin x(1)=x/square(1-x^2) +arcsin x
Should you check your solutions for absolute values for derivatives
Yes
Find inverse function algebraically
You have f(x)=(x-1)/(x+1) y=(x-1)/(x+1) x=(y-1)(y+2) times (y+2) to get rid of (y+2) so you have xy+2x=(y-1) -y xy+2x-y=1 -2x xy-y=-2x-1 isolate y terms y(x-1)=-2x-1 divide (x-1) f of inverse = (-2x-1)/(x-1) check to see if it is a function with vertical line test
How to find the minimum and solve a quadratic function f(x)=2x^2-8x+1
You need to take the derivative to get the minimum Algerialy f'(x)=4x-8 x=8/4 x=2 min (2, f(2) (f the original function) )) min (2, -7) f is decreasing (- infinity, 2) and increasing (2, infinity)
log b ^x how to solve
a log must equal an exponent b is the base the base with the exponent equals x
Definition of critical numbers
a number a in the domain of a given function f is called a critical number of f if f'(a)=0 or f' is underfined at x=a
how to solve derivative for basic absolute value f(x)=abs (x-1)
absolute val u=square root of u^2 which is y u=x-1 which is x f(x)=y=square toot of u^2 use chain rule f'(x)=(dy/du) times (dx/du) f'(x)=1/2(2u) times square root of u^2(du/dx) u times u'/absolute value of u f'(x)= u times 1/square root of (u^2)=(x-1)/absolute value of x-1 when x is greater then 0 f'(x)=1 when x is less than o f'(x)=-1 which does not exist at x=1
derivative for sinx
cos x
derivative for sinh x
cosh x
sum to product formulas for cos cos
cosx + cosy=2 times cos((x+y)/2 times cos ((x-y)/2 cosx -cos y=2 times sin((x+y/2) times sin (x-y)/2)
product to sum formulas for cos cos
cosx cosy=1/2(cos(x+y) +cos(x-y))
product to sum formulas for cos sin
cosx siny=1/2(sin(x+y)-sin(x-y))
u and y in the chain rule stands for
dx=u=x y is y
Implicit differentiation equation
dy/dx
derivative for e^x
e^x
the equation for the linear approximation of functions
f l(x)=f(a) + f'(a)(x-a)
inverse derivative of arccos x
f'(x)= -1/square root of (1-x^2)
defintion to find derivative equation
f'(x)= lim (this is under lim) h to 0 (f(x+h))/h
How to solve critical numbers (x^2 +7)/(x+3)
f'(x)=(2x(x+3) -x^2+7)(1)/(x+3)^2) f'(x)=(x^2+6x-7)/(x+3)^2 0=x^2+6x-7 (x+7)(x-1)=0 x=-7 or x=1 and they are critical numbers
chain rule for derivatives equation
f'(x)=(dy/du) times (du/dx)
inverse derivative of arccot x
f'(x)=-1/(1+x^2)
inverse derivative of arccsc x
f'(x)=-1/(x square root of (x^2-1
derivative equation for hyperbolic function of f(x)=csch x or 1/sinh x
f'(x)=-csch x coth x
derivative equation for hyperbolic function of f(x)=sech x or 1/cosh x
f'(x)=-sech x tanh x
How to solve differentiation of logarithmic function f(x)= log 3 (3 is under log) x
f'(x)=1(x ln 3)
inverse derivative of arctan x
f'(x)=1/(1+x^2)
When f(x)=log b (b is under log) x derivative equation for differentiation of exponential function
f'(x)=1/(x ln b)
inverse derivative of arcsin x
f'(x)=1/square root of (1-x^2)
inverse derivative of arcsec x
f'(x)=1/x square root of (x^2-1)
How to solve differentiation of exponential function f(x)=2^x
f'(x)=2^x ln 2
Derivative equation for differentiation of exponential functions when it is f(x)=b^x
f'(x)=b^x ln b
derivative equation for hyperbolic function of f(x)=sinh x
f'(x)=cosh x
derivative equation for hyperbolic function of f(x)=coth x or 1/tanh x or cosh x/sinh x
f'(x)=csch ^2 x
A derivative that shows that arcsin(x) + arccos(x)=pie/2
f'(x)=d( arcsin(x)/dx + d(arccos(x)/dx f'(x)=1/square root of (1-x^2) + (-1/square root of (1-x^2) f'(x)=0 and how to prove it f(0)=arcsin(0)+ arccos(1)= 0+pie/2 f(1)=arcsin(1) + arccso(1)=pie/2+0
derivative equation for hyperbolic function of f(x)=tanh x or sinhx/coshx
f'(x)=sech^2 x
derivative equation for hyperbolic function of f(x)=cosh x
f'(x)=sinh x
Standard vertex form for quadratic functions
f(x)=a (x-h)^2
Derivation of a function times by a constant
f(x)=c g(x) f'(x)=c g'(x) f(x)=3x^3 c=3 g(x)=x^3 f'(x)=3(3x^2)=9x^2
derivative for f(x)
f(x)=d(F(x))/dx) du/dx
derivative of the product of two functions product rule
f(x)=g(x) times h(x) f'(x)=g(x) times h'(x) + h(x) times g'(x) f(x)=(x^2-2x)(x-2) f'(x)=(x^2-2x)(1)+(x-2)(2x-2)
Derivative of the sum of functions sum rule
f(x)=g(x)+h(x) f'(x)=g'(x)+h'(x) f(x)=x^2+4 g(x)=x^2 h(x)=4 f'(x)=2x+0=2x
derivative of the difference of functions
f(x)=g(x)-h(x) f'(x)=g'(x)-h'(x) f(x)=x^3-x^-2 g(x)=x^3 h(x)=-x^-2 f'(x)=3x^2-(-2x^-3)=3x^2+2x^-3
derivative of the quotient of two functions quotient rule
f(x)=g(x)/h(x) f'(x)=(h(x) g'(x) - g(x) h'(x) )/h(x)^2 f(x)=(x-2)/(x+1) f'(x)= ((x+1)(1) -(x-2)(1))-(x+1)^2= 3/(x+1)^2
Derivation of a power function power rule
f(x)=x^r f'(x)=rx^r-1 applies with all product rules of derivates when their is a power f(x)=3x^2 f'(x)=3x
How to solve vertex form for quadratic functions f(x)=-(x+3)^2+1
find the derivative f'(x)=-2(x-3) maximum at (-3,1) f increase on (-infinity,-3) and decrease on (-3, infinity)
How to solve for zero with the newton method f(x)=x^2+3x+1
find zeros first then find the derivative start with zero or a value close to it x1=x0-f(x0)/f'(x0) x1=0-f(0)/f'(0) x1=-(0^2+0x+1)/(2(0)+3) x1=-1/3 keep on putting in values until close to zero and plug it back in the original equation
How to solve linear approximation of functions f(x)=ln x for x close to one
first compute f'(1) f'(x)=1/x f'(1)=1 f l (x)=ln1 +f'(1)(x-1)=x-1 the results means ln x is about x-1 for x close to one use calculator to calculate ln x and x-1 for x=1 x=1.001 x=1.01 x=1.1 x=1.5 Compare ln x and x-1
How to solve the derivative of inverse function f(x)=(1/2)x-1
get the inverse of y other method f(f^-1(x))=x f(y)=x (dy/dx)(df/dy)=1 this is the chain rule dy/dx=1/(df/dy) (dy/dx)=1/(1/2)=2 x=2
how to solve chain rule problem for f(x)=(x^3-4x+5)^4
let u=(x^3-4x+5) y=u^4 du/dx=3x^2-4 dy/du=4u^3 du/dx 4x to 4 from product form of consont equation dy/du from one of the product form equation times using the chain rule and substitute u for the function where it equals u
natural log form
ln e^5=x ln is the e known as the natural log it is the base too e^5 is what your are solving for and x is the exponent the base and exponent are together
how to solve logarithmic differentiation of y=3x^2e^-x
ln y= ln 3 + ln x^2 +ln ^e^-x ln y= ln 3+ 2 lnx -x y'/y=0+2/x-1 multiply all terms by y
logarithmic form
log b ^x=y y is the exponent b is the base x is what you solving for the base and exponent are together when solving for x
Product property of log
log b xy=logb X +logb Y
Power Property of log
logb X^p= Plogb X
Quotient Property of log
logb x/y= logb X- logb Y
difference quotient and slope equation and examples
m=(f(x+h)-f(x)/h f(x)=2x+5 f(x+h)=2(x+h)+5 ( f(x-h) -f(x)) /h= ( 2(x+h) +5 -(2x+5) )/h= 2h/h=2
derivative for x^n
n x^n-1
derivative for tanx
sec^2x
derivative for tanh x
sech^2 x
derivative for secx
secx tanx
derivative for cosh x
sinh x
sum to product formulas for sin sin
sinx + siny=2 times sin(x+y)/2) times cos(x-y)/2) sinx-siny= 2 times cos ((x+y)/2) times sin((x-y)/2)
product to sum formulas for sin cos
sinx cos y=1/2(sin(x+y) +sin(x-y))
product to sum formulas for sin sin
sinx siny=1/2(cos(x-y)-cos(x+y))
graph of arccos(cos(x))
the regular graph of cosx has a continous loop while arccos graph starts at (0,0) and (pie, pie) then (0,2pie)
graph of arcsin(sin(x))
the regular graph of sinx starts at (0,0) and then loops to (pie,pie) and then loops to (2pie,2pie) so the arcsin graph starts at (0,0) and then at (pie/2,pie/2) and then crosses to (0,-pie/2)
graph of arctan(tan(x))
the regular graph of tanx has the first positive vertical asstompye at pie/2 and negative first one at -pie/2 and for the arctan graph it crosses right in the middle of the curved graph of tanx
How to find the sum of absolute value and its derivative f(x)=-x+2+ absolute val -x+2
u=-x+2 f'(x)=-1+u u'/absolute val u=-1+(-x+2)(-1)/absolute value of -x+2 x is less then 2 and f'(x)=2 if x is greater than 2 f'(x)=0 x=2 does not exist
How to solve implicit differentiation with derivative where y^4+xy^2=3
use product form for sums d(y^4)/dx+d(xy^4)/dx+d(x)/dx=d(3)/dx use power rule, product and chain rule 4y^3dy/dx+(1)y^2+x2y dy/dx +1=0 dy/dx=(-1-y^2)/(4y^3+2xy)
How to solve differiation of hyperbolic function f(x)=sinh (x^2)
use this equation (dy/du) times (du/dx) u=x^2 y=sinh u f'(x)=2x cosh u x^2=2x substitute u for x^2 and f'(x)=2x cosh x^2
how to solve defintion to find derivative problem like f(x)= mx+b
used the difference quotient equation and you get (mh)/h=m the limit is h to 0 so the derivative is f'(x)=m
product rule of a derivative of a constant function
when f(x)=-10 then f'(x)=0
Newton Method to find zeros equation
x n+1 (n+1 is under x)=x n (n is under x) - f Xn (x is under f and n is under x) /f'(Xn) n=1,2,3 and so on
How to solve critical numbers of f(x)= absolute value of x-2
y=square root u^2 u=dx= x-2 using chain rule f'(x)=(1/2)2 u u'/ abs u f'(x)=(x-2)/abs(x-2) f' is underfined at x=2 x=-2 x=2 is the critical nunber