حمّل تطبيق منهاجي الجديد

منهاجي صار أسرع من خلال التطبيق

  أتدرب وأحل المسائل

أتدرب وأحل المسائل

مشتقتا الاقتران الأسي الطبيعي والاقتران اللوغاريتمي الطبيعي

أجد مشتقة كل اقتران ممّا يأتي:

(1) begin mathsize 20px style f left parenthesis x right parenthesis equals 2 e to the power of x plus 1 end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 2 e to the power of x end style

(2) begin mathsize 20px style f left parenthesis x right parenthesis equals e to the power of 3 x plus 9 end exponent end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 3 e to the power of 3 x plus 9 end exponent end style

(3) begin mathsize 20px style f left parenthesis x right parenthesis equals left parenthesis x squared plus 3 x minus 9 right parenthesis e to the power of x end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis x squared plus 3 x minus 9 right parenthesis left parenthesis e to the power of x right parenthesis plus left parenthesis e to the power of x right parenthesis left parenthesis 2 x plus 3 right parenthesis equals e to the power of x left parenthesis x squared plus 5 x minus 6 right parenthesis end style

(4) begin mathsize 20px style f left parenthesis x right parenthesis equals e to the power of x over x to the power of 4 end style

fbegin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator x to the power of 4 e to the power of x minus e to the power of x left parenthesis 4 x cubed right parenthesis over denominator x to the power of 8 end fraction equals fraction numerator x e to the power of x minus 4 e to the power of x over denominator x to the power of 5 end fraction end style

(5) begin mathsize 20px style f left parenthesis x right parenthesis equals 6 e to the power of square root of x end exponent end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 6 cross times fraction numerator 1 over denominator 2 square root of x end fraction e to the power of square root of x end exponent equals fraction numerator 3 over denominator square root of x end fraction e to the power of square root of x end exponent end style

(6) begin mathsize 20px style f left parenthesis x right parenthesis equals fraction numerator e to the power of x over denominator 1 plus e to the power of x end fraction end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator left parenthesis 1 plus e to the power of x right parenthesis left parenthesis e to the power of x right parenthesis minus e to the power of x left parenthesis e to the power of x right parenthesis over denominator left parenthesis 1 plus e to the power of x right parenthesis squared end fraction equals e to the power of x over left parenthesis 1 plus e to the power of x right parenthesis squared end style

(7) begin mathsize 20px style f left parenthesis x right parenthesis equals left parenthesis e to the power of x plus 2 right parenthesis left parenthesis e to the power of x minus 1 right parenthesis end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis e to the power of x plus 2 right parenthesis left parenthesis e to the power of x right parenthesis plus left parenthesis e to the power of x minus 1 right parenthesis left parenthesis e to the power of x right parenthesis equals 2 e to the power of 2 x end exponent plus e to the power of x end style

(8) begin mathsize 20px style f left parenthesis x right parenthesis equals e to the power of negative 2 x end exponent left parenthesis 2 x minus 1 right parenthesis to the power of 5 end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals left parenthesis e to the power of negative 2 x end exponent right parenthesis cross times 5 left parenthesis 2 x minus 1 right parenthesis to the power of 4 cross times 2 plus left parenthesis 2 x minus 1 right parenthesis to the power of 5 left parenthesis negative 2 e to the power of negative 2 x end exponent right parenthesis end cell row blank cell equals 2 e to the power of negative 2 x end exponent left parenthesis 2 x minus 1 right parenthesis to the power of 4 left parenthesis 6 minus 2 x right parenthesis end cell end table end style

(9) begin mathsize 20px style f left parenthesis x right parenthesis equals x cubed minus 5 e to the power of 2 x end exponent end style

fbegin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 3 x squared minus 5 cross times 2 e to the power of 2 x end exponent equals 3 x squared minus 10 e to the power of 2 x end exponent end style

 

أجد مشتقة كل اقتران ممّا يأتي:

(10) begin mathsize 20px style f left parenthesis x right parenthesis equals 3 ln invisible function application space x end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 3 over x end style

(11) begin mathsize 20px style f left parenthesis x right parenthesis equals x cubed ln invisible function application space x end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis x cubed right parenthesis left parenthesis 1 over x right parenthesis plus left parenthesis l n invisible function application x right parenthesis left parenthesis 3 x squared right parenthesis equals x squared plus 3 x squared l n invisible function application x end style

(12) begin mathsize 20px style f left parenthesis x right parenthesis equals fraction numerator ln invisible function application space x over denominator x squared end fraction end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator x squared left parenthesis 1 over x right parenthesis minus left parenthesis l n invisible function application x right parenthesis left parenthesis 2 x right parenthesis over denominator x to the power of 4 end fraction equals fraction numerator x minus 2 x l n invisible function application x over denominator x to the power of 4 end fraction equals fraction numerator 1 minus 2 l n invisible function application x over denominator x cubed end fraction end style

(13) begin mathsize 20px style f left parenthesis x right parenthesis equals x squared ln invisible function application left parenthesis 4 x right parenthesis end style

fbegin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis x squared right parenthesis left parenthesis fraction numerator 4 over denominator 4 x end fraction right parenthesis plus left parenthesis l n invisible function application left parenthesis 4 x right parenthesis right parenthesis left parenthesis 2 x right parenthesis equals x plus 2 x l n invisible function application left parenthesis 4 x right parenthesis end style

(14) begin mathsize 20px style f left parenthesis x right parenthesis equals ln invisible function application left parenthesis fraction numerator x plus 1 over denominator x end fraction right parenthesis end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator fraction numerator left parenthesis x right parenthesis left parenthesis 1 right parenthesis minus left parenthesis x plus 1 right parenthesis left parenthesis 1 right parenthesis over denominator x squared end fraction over denominator fraction numerator x plus 1 over denominator x end fraction end fraction equals fraction numerator fraction numerator negative 1 over denominator x squared end fraction over denominator fraction numerator x plus 1 over denominator x end fraction end fraction equals fraction numerator negative 1 over denominator x squared end fraction cross times fraction numerator x over denominator x plus 1 end fraction equals fraction numerator negative 1 over denominator x left parenthesis x plus 1 right parenthesis end fraction end style

(15) begin mathsize 20px style f left parenthesis x right parenthesis equals ln invisible function application square root of x squared minus 1 end root end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator fraction numerator 2 x over denominator 2 square root of x squared minus 1 end root end fraction over denominator square root of x squared minus 1 end root end fraction equals fraction numerator 2 x over denominator 2 square root of x squared minus 1 end root end fraction cross times fraction numerator 1 over denominator square root of x squared minus 1 end root end fraction equals fraction numerator x over denominator x squared minus 1 end fraction end style

(16) fbegin mathsize 20px style f left parenthesis x right parenthesis equals left parenthesis ln invisible function application x right parenthesis to the power of 4 end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 4 left parenthesis l n invisible function application x right parenthesis cubed cross times 1 over x equals fraction numerator 4 left parenthesis l n invisible function application x right parenthesis cubed over denominator x end fraction end style

(17) begin mathsize 20px style f left parenthesis x right parenthesis equals ln invisible function application left parenthesis x squared minus 5 right parenthesis end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x squared minus 5 end fraction end style

(18) begin mathsize 20px style f left parenthesis x right parenthesis equals x to the power of 4 ln invisible function application x minus 1 half e to the power of x end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis x to the power of 4 right parenthesis left parenthesis 1 over x right parenthesis plus left parenthesis l n invisible function application x right parenthesis left parenthesis 4 x cubed right parenthesis minus 1 half e to the power of x equals x cubed plus 4 x cubed l n invisible function application x minus 1 half e to the power of x end style

(19) begin mathsize 20px style f left parenthesis x right parenthesis equals e to the power of 2 x end exponent ln invisible function application x end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis e to the power of 2 x end exponent right parenthesis left parenthesis 1 over x right parenthesis plus left parenthesis l n invisible function application x right parenthesis left parenthesis 2 e to the power of 2 x end exponent right parenthesis equals fraction numerator e to the power of 2 x end exponent left parenthesis 1 plus x l n invisible function application x right parenthesis over denominator x end fraction end style

(20) fbegin mathsize 20px style f left parenthesis x right parenthesis equals left parenthesis ln invisible function application 3 x right parenthesis left parenthesis ln invisible function application 7 x right parenthesis end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis l n invisible function application 3 x right parenthesis left parenthesis fraction numerator 7 over denominator 7 x end fraction right parenthesis plus left parenthesis l n invisible function application 7 x right parenthesis left parenthesis fraction numerator 3 over denominator 3 x end fraction right parenthesis equals fraction numerator l n invisible function application 3 x plus l n invisible function application 7 x over denominator x end fraction end style

(21) begin mathsize 20px style f left parenthesis x right parenthesis equals ln invisible function application left parenthesis e to the power of x minus 2 right parenthesis end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals fraction numerator e to the power of x over denominator e to the power of x minus 2 end fraction end style

 

أجد مشتقة كل اقتران ممّا يأتي عند قيمة x المعطاة:

(22) begin mathsize 20px style f left parenthesis x right parenthesis equals e to the power of 2 x minus 1 end exponent ln invisible function application left parenthesis 2 x minus 1 right parenthesis space comma space x equals 1 end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f to the power of straight prime left parenthesis x right parenthesis equals left parenthesis e to the power of 2 x minus 1 end exponent right parenthesis left parenthesis fraction numerator 2 over denominator 2 x minus 1 end fraction right parenthesis plus left parenthesis l n invisible function application left parenthesis 2 x minus 1 right parenthesis right parenthesis left parenthesis 2 e to the power of 2 x minus 1 end exponent right parenthesis end cell row blank cell f to the power of straight prime left parenthesis 1 right parenthesis equals left parenthesis e to the power of 2 minus 1 end exponent right parenthesis left parenthesis fraction numerator 2 over denominator 2 minus 1 end fraction right parenthesis plus left parenthesis l n invisible function application left parenthesis 2 minus 1 right parenthesis right parenthesis left parenthesis 2 e to the power of 2 minus 1 end exponent right parenthesis equals 2 e plus 0 equals 2 e end cell end table end style

 

(23) begin mathsize 20px style f left parenthesis x right parenthesis equals fraction numerator ln invisible function application x squared over denominator x end fraction space comma space x equals 4 end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals fraction numerator x left parenthesis fraction numerator 2 x over denominator x squared end fraction right parenthesis minus left parenthesis l n invisible function application x squared right parenthesis left parenthesis 1 right parenthesis over denominator x squared end fraction equals fraction numerator 2 minus l n invisible function application x squared over denominator x squared end fraction end cell row cell f to the power of straight prime left parenthesis 4 right parenthesis end cell cell equals fraction numerator 2 minus l n invisible function application 16 over denominator 16 end fraction end cell end table end style

 

(24) فيروسات: يمكن نمذجة انتشار الإنفلونزا في إحدى المدارس باستعمال الاقتران: begin mathsize 20px style P left parenthesis t right parenthesis equals fraction numerator 100 over denominator 1 plus e to the power of 3 minus t end exponent end fraction end style ، حيث P(t) عدد السنوات بعد t يوماً من ملاحظة الإنفلونزا أوّل مرّة في المدرسة. أجد سرعة انتشار الإنفلونزا في المدرسة بعد 3 أيام.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell P to the power of straight prime left parenthesis t right parenthesis equals fraction numerator negative 100 cross times negative e to the power of 3 minus t end exponent over denominator left parenthesis 1 plus e to the power of 3 minus t end exponent right parenthesis squared end fraction equals fraction numerator 100 e to the power of 3 minus t end exponent over denominator left parenthesis 1 plus e to the power of 3 minus t end exponent right parenthesis squared end fraction end cell row blank cell P to the power of straight prime left parenthesis 3 right parenthesis equals fraction numerator 100 e to the power of 3 minus 3 end exponent over denominator left parenthesis 1 plus e to the power of 3 minus 3 end exponent right parenthesis squared end fraction equals 100 over 4 equals 25 end cell end table end style

 

(25) ذاكرة: يُستعمل الاقتران: begin mathsize 20px style m left parenthesis t right parenthesis equals t ln invisible function application space t plus 1 space comma space 0 less than t less or equal than 4 end style لقياس قدرة الأطفال على التذكر، حيث m مقياس من 1 إلى 7 ، و t عمر الطفل بالسنوات. أجد معدل تغير قدرة الأطفال على التذكر بالنسبة إلى عمر الطفل t .

begin mathsize 20px style m to the power of straight prime left parenthesis t right parenthesis equals left parenthesis t right parenthesis left parenthesis 1 over t right parenthesis plus left parenthesis l n invisible function application space t right parenthesis left parenthesis 1 right parenthesis equals 1 plus l n invisible function application space t end style

 

أستعمل قاعدة السلسلة في إيجاد begin mathsize 20px style fraction numerator d y over denominator d x end fraction end style لكلّ ممّا يأتي:

(26) begin mathsize 20px style y equals e to the power of 2 u end exponent plus 3 space comma space u equals x squared plus 1 end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator d y over denominator d u end fraction end cell cell equals 2 e to the power of 2 u end exponent end cell row cell fraction numerator d u over denominator d x end fraction end cell cell equals 2 x end cell row cell fraction numerator d y over denominator d x end fraction end cell cell equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction end cell row blank cell equals 2 e to the power of 2 u end exponent cross times 2 x end cell row blank cell equals 4 x e to the power of 2 u end exponent end cell row blank cell equals 4 x e to the power of 2 left parenthesis x squared plus 1 right parenthesis end exponent end cell end table end style

 

(27) begin mathsize 20px style y equals ln invisible function application left parenthesis u plus 1 right parenthesis space comma space u equals e to the power of x end style

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator d y over denominator d u end fraction end cell cell equals fraction numerator 1 over denominator u plus 1 end fraction end cell row cell fraction numerator d u over denominator d x end fraction end cell cell equals e to the power of x end cell row cell fraction numerator d y over denominator d x end fraction end cell cell equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction end cell row blank cell equals fraction numerator 1 over denominator u plus 1 end fraction cross times e to the power of x end cell row blank cell equals fraction numerator e to the power of x over denominator e to the power of x plus 1 end fraction end cell end table end style

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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