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We can use a formula to find the derivative ofy=lnx,y=lnx,and the relationshiplogbx=lnxlnblogbx=lnxlnballows us to extend our differentiation formulas to include logarithms with arbitrary bases. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
Logarithmic differentiation allows us to differentiate functions of the formy=g(x)f(x)y=g(x)f(x)or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating. | https://openstax.org/books/calculus-volume-1/pages/3-key-concepts |
Q = f ( x ) â f ( a ) x â a Q = f ( x ) â f ( a ) x â a | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
Q = f ( a + h ) â f ( a ) a + h â a = f ( a + h ) â f ( a ) h Q = f ( a + h ) â f ( a ) a + h â a = f ( a + h ) â f ( a ) h | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
m tan = lim x â a f ( x ) â f ( a ) x â a m tan = lim x â a f ( x ) â f ( a ) x â a m tan = lim h â 0 f ( a + h ) â f ( a ) h m tan = lim h â 0 f ( a + h ) â f ( a ) h | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
f â² ( a ) = lim x â a f ( x ) â f ( a ) x â a f â² ( a ) = lim x â a f ( x ) â f ( a ) x â a f â² ( a ) = lim h â 0 f ( a + h ) â f ( a ) h f â² ( a ) = lim h â 0 f ( a + h ) â f ( a ) h | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
v a ve = s ( t ) â s ( a ) t â a v a ve = s ( t ) â s ( a ) t â a | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
v ( a ) = s â² ( a ) = lim t â a s ( t ) â s ( a ) t â a v ( a ) = s â² ( a ) = lim t â a s ( t ) â s ( a ) t â a | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
f â² ( x ) = lim h â 0 f ( x + h ) â f ( x ) h f â² ( x ) = lim h â 0 f ( x + h ) â f ( x ) h | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( sin x ) = cos x d d x ( sin x ) = cos x | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( cos x ) = â sin x d d x ( cos x ) = â sin x | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( tan x ) = sec 2 x d d x ( tan x ) = sec 2 x | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( cot x ) = â csc 2 x d d x ( cot x ) = â csc 2 x | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( sec x ) = sec x tan x d d x ( sec x ) = sec x tan x | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( csc x ) = â csc x cot x d d x ( csc x ) = â csc x cot x | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
h â² ( x ) = f â² ( g ( x ) ) g â² ( x ) h â² ( x ) = f â² ( g ( x ) ) g â² ( x ) | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
h â² ( x ) = n ( g ( x ) ) n â 1 g â² ( x ) h â² ( x ) = n ( g ( x ) ) n â 1 g â² ( x ) | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
( f â1 ) â² ( x ) = 1 f â² ( f â1 ( x ) ) ( f â1 ) â² ( x ) = 1 f â² ( f â1 ( x ) ) whenever f â² ( f â1 ( x ) ) â 0 f â² ( f â1 ( x ) ) â 0 and f ( x ) f ( x ) is differentiable. | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( x m / n ) = m n x ( m / n ) â 1 . d d x ( x m / n ) = m n x ( m / n ) â 1 . | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x sin â1 x = 1 1 â ( x ) 2 d d x sin â1 x = 1 1 â ( x ) 2 | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x cos â1 x = â1 1 â ( x ) 2 d d x cos â1 x = â1 1 â ( x ) 2 | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x tan â1 x = 1 1 + ( x ) 2 d d x tan â1 x = 1 1 + ( x ) 2 | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x cot â1 x = â1 1 + ( x ) 2 d d x cot â1 x = â1 1 + ( x ) 2 | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x sec â1 x = 1 | x | ( x ) 2 â 1 d d x sec â1 x = 1 | x | ( x ) 2 â 1 | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x csc â1 x = â1 | x | ( x ) 2 â 1 d d x csc â1 x = â1 | x | ( x ) 2 â 1 | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( e g ( x ) ) = e g ( x ) g â² ( x ) d d x ( e g ( x ) ) = e g ( x ) g â² ( x ) | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( ln g ( x ) ) = 1 g ( x ) g â² ( x ) d d x ( ln g ( x ) ) = 1 g ( x ) g â² ( x ) | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( b g ( x ) ) = b g ( x ) g â² ( x ) ln b d d x ( b g ( x ) ) = b g ( x ) g â² ( x ) ln b | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
d d x ( log b g ( x ) ) = g â² ( x ) g ( x ) ln b d d x ( log b g ( x ) ) = g â² ( x ) g ( x ) ln b | https://openstax.org/books/calculus-volume-1/pages/3-key-equations |
acceleration : is the rate of change of the velocity, that is, the derivative of velocity | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
amount of change : the amount of a functionf(x)f(x)over an interval[x,x+h][x,x+h]isf(x+h)âf(x)f(x+h)âf(x) | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
average rate of change : is a functionf(x)f(x)over an interval[x,x+h][x,x+h]isf(a+h)âf(a)hf(a+h)âf(a)h | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
chain rule : the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
constant multiple rule : the derivative of a constantcmultiplied by a functionfis the same as the constant multiplied by the derivative:ddx(cf(x))=cfâ²(x)ddx(cf(x))=cfâ²(x) | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
constant rule : the derivative of a constant function is zero:ddx(c)=0,ddx(c)=0,wherecis a constant | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
derivative : the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
derivative function : gives the derivative of a function at each point in the domain of the original function for which the derivative is defined | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
difference quotient : of a functionf(x)f(x)ataais given byf(a+h)âf(a)horf(x)âf(a)xâaf(a+h)âf(a)horf(x)âf(a)xâa | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
difference rule : the derivative of the difference of a functionfand a functiongis the same as the difference of the derivative offand the derivative ofg:ddx(f(x)âg(x))=fâ²(x)âgâ²(x)ddx(f(x)âg(x))=fâ²(x)âgâ²(x) | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
differentiable ata : a function for whichfâ²(a)fâ²(a)exists is differentiable ataa | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
differentiable function : a function for whichfâ²(x)fâ²(x)exists is a differentiable function | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
differentiable onS : a function for whichfâ²(x)fâ²(x)exists for eachxxin the open setSSis differentiable onSS | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
differentiation : the process of taking a derivative | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
higher-order derivative : a derivative of a derivative, from the second derivative to thenth derivative, is called a higher-order derivative | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
implicit differentiation : is a technique for computingdydxdydxfor a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variableyyas a function) and solving fordydxdydx | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
instantaneous rate of change : the rate of change of a function at any point along the functiona,a,also calledfâ²(a),fâ²(a),or the derivative of the function ataa | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
logarithmic differentiation : is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
marginal cost : is the derivative of the cost function, or the approximate cost of producing one more item | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
marginal profit : is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
marginal revenue : is the derivative of the revenue function, or the approximate revenue obtained by selling one more item | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
population growth rate : is the derivative of the population with respect to time | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
power rule : the derivative of a power function is a function in which the power onxxbecomes the coefficient of the term and the power onxxin the derivative decreases by 1: Ifnnis an integer, thenddxxn=nxnâ1ddxxn=nxnâ1 | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
product rule : the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function:ddx(f(x)g(x))=fâ²(x)g(x)+gâ²(x)f(x)ddx(f(x)g(x))=fâ²(x)g(x)+gâ²(x)f(x) | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
quotient rule : the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function:ddx(f(x)g(x))=fâ²(x)g(x)âgâ²(x)f(x)(g(x))2ddx(f(x)g(x))=fâ²(x)g(x)... | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
speed : is the absolute value of velocity, that is,|v(t)||v(t)|is the speed of an object at timettwhose velocity is given byv(t)v(t) | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
sum rule : the derivative of the sum of a functionfand a functiongis the same as the sum of the derivative offand the derivative ofg:ddx(f(x)+g(x))=fâ²(x)+gâ²(x)ddx(f(x)+g(x))=fâ²(x)+gâ²(x) | https://openstax.org/books/calculus-volume-1/pages/3-key-terms |
To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
In terms of the quantities, state the information given and the rate to be found. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Find an equation relating the quantities. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
A differentiable functiony=f(x)y=f(x)can be approximated ataaby the linear functionL(x)=f(a)+fâ²(a)(xâa).L(x)=f(a)+fâ²(a)(xâa). | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
For a functiony=f(x),y=f(x),ifxxchanges fromaatoa+dx,a+dx,thendy=fâ²(x)dxdy=fâ²(x)dxis an approximation for the change iny.y.The actual change inyyisÎy=f(a+dx)âf(a).Îy=f(a+dx)âf(a). | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
A measurement errordxdxcan lead to an error in a calculated quantityf(x).f(x).The error in the calculated quantity is known as thepropagated error. The propagated error can be estimated bydyâfâ²(x)dx.dyâfâ²(x)dx. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
To estimate the relative error of a particular quantityq,q,we estimateÎqq.Îqq. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Ifffis continuous over[a,b][a,b]and differentiable over(a,b)(a,b)andf(a)=f(b),f(a)=f(b),then there exists a pointcâ(a,b)câ(a,b)such thatfâ²(c)=0.fâ²(c)=0.This is Rolleâs theorem. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Ifffis continuous over[a,b][a,b]and differentiable over(a,b),(a,b),then there exists a pointcâ(a,b)câ(a,b)such thatfâ²(c)=f(b)âf(a)bâa.fâ²(c)=f(b)âf(a)bâa.This is the Mean Value Theorem. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ²(x)=0fâ²(x)=0over an intervalI,I,thenffis constant overI.I. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
If two differentiable functionsffandggsatisfyfâ²(x)=gâ²(x)fâ²(x)=gâ²(x)overI,I,thenf(x)=g(x)+Cf(x)=g(x)+Cfor some constantC.C. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ²(x)>0fâ²(x)>0over an intervalI,I,thenffis increasing overI.I.Iffâ²(x)<0fâ²(x)<0overI,I,thenffis decreasing overI.I. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Ifccis a critical point offfandfâ²(x)>0fâ²(x)>0forx<cx<candfâ²(x)<0fâ²(x)<0forx>c,x>c,thenffhas a local maximum atc.c. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Ifccis a critical point offfandfâ²(x)<0fâ²(x)<0forx<cx<candfâ²(x)>0fâ²(x)>0forx>c,x>c,thenffhas a local minimum atc.c. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ³(x)>0fâ³(x)>0over an intervalI,I,thenffis concave up overI.I. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ³(x)<0fâ³(x)<0over an intervalI,I,thenffis concave down overI.I. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ²(c)=0fâ²(c)=0andfâ³(c)>0,fâ³(c)>0,thenffhas a local minimum atc.c. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ²(c)=0fâ²(c)=0andfâ³(c)<0,fâ³(c)<0,thenffhas a local maximum atc.c. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Iffâ²(c)=0fâ²(c)=0andfâ³(c)=0,fâ³(c)=0,then evaluatefâ²(x)fâ²(x)at a test pointxxto the left ofccand a test pointxxto the right ofc,c,to determine whetherffhas a local extremum atc.c. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
The limit off(x)f(x)isLLasxââxââ(or asxâââ)xâââ)if the valuesf(x)f(x)become arbitrarily close toLLasxxbecomes sufficiently large. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
The limit off(x)f(x)isââasxââxââiff(x)f(x)becomes arbitrarily large asxxbecomes sufficiently large. The limit off(x)f(x)isââââasxââxââiff(x)<0f(x)<0and|f(x)||f(x)|becomes arbitrarily large asxxbecomes sufficiently large. We can define the limit off(x)f(x)asxxapproachesââââsimilarly. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
For a polynomial functionp(x)=anxn+anâ1xnâ1+â¦+a1x+a0,p(x)=anxn+anâ1xnâ1+â¦+a1x+a0,whereanâ0,anâ0,the end behavior is determined by the leading termanxn.anxn.Ifnâ0,nâ0,p(x)p(x)approachesââorââââat each end. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
For a rational functionf(x)=p(x)q(x),f(x)=p(x)q(x),the end behavior is determined by the relationship between the degree ofppand the degree ofq.q.If the degree ofppis less than the degree ofq,q,the liney=0y=0is a horizontal asymptote forf.f.If the degree ofppis equal to the degree ofq,q,then the liney=anbny=anbnis a ho... | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
To solve an optimization problem, begin by drawing a picture and introducing variables. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Find an equation relating the variables. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Find a function of one variable to describe the quantity that is to be minimized or maximized. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Look for critical points to locate local extrema. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
LâHôpitalâs rule can be used to evaluate the limit of a quotient when the indeterminate form0000orâ/ââ/âarises. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
LâHôpitalâs rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form0000orâ/â.â/â. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
The exponential functionexexgrows faster than any power functionxp,xp,p>0.p>0. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
The logarithmic functionlnxlnxgrows more slowly than any power functionxp,xp,p>0.p>0. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Newtonâs method approximates roots off(x)=0f(x)=0by starting with an initial approximationx0,x0,then uses tangent lines to the graph offfto create a sequence of approximationsx1,x2,x3,â¦.x1,x2,x3,â¦. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Typically, Newtonâs method is an efficient method for finding a particular root. In certain cases, Newtonâs method fails to work because the list of numbersx0,x1,x2,â¦x0,x1,x2,â¦does not approach a finite value or it approaches a value other than the root sought. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Any process in which a list of numbersx0,x1,x2,â¦x0,x1,x2,â¦is generated by defining an initial numberx0x0and defining the subsequent numbers by the equationxn=F(xnâ1)xn=F(xnâ1)for some functionFFis an iterative process. Newtonâs method is an example of an iterative process, where the functionF(x)=xâ[f(x)fâ²... | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
IfFFis an antiderivative off,f,then every antiderivative offfis of the formF(x)+CF(x)+Cfor some constantC.C. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
Solving the initial-value problemdydx=f(x),y(x0)=y0dydx=f(x),y(x0)=y0requires us first to find the set of antiderivatives offfand then to look for the particular antiderivative that also satisfies the initial condition. | https://openstax.org/books/calculus-volume-1/pages/4-key-concepts |
L ( x ) = f ( a ) + f â² ( a ) ( x â a ) L ( x ) = f ( a ) + f â² ( a ) ( x â a ) | https://openstax.org/books/calculus-volume-1/pages/4-key-equations |
d y = f â² ( x ) d x . d y = f â² ( x ) d x . | https://openstax.org/books/calculus-volume-1/pages/4-key-equations |
absolute extremum : ifffhas an absolute maximum or absolute minimum atc,c,we sayffhas an absolute extremum atcc | https://openstax.org/books/calculus-volume-1/pages/4-key-terms |
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