حمّل تطبيق منهاجي الجديد

منهاجي صار أسرع من خلال التطبيق

  أتحقق من فهمي

أتحقق من فهمي

مشتقتا الاقتران الأسي الطبيعي والاقتران اللوغاريتمي الطبيعي

مشتقة الاقتران الأسي الطبيعي

أتحقق من فهمي صفحة 74

أجد مشتقة كل اقتران ممّا يأتي:

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

(b) begin mathsize 20px style f left parenthesis x right parenthesis equals cube root of x plus 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 1 third x to the power of negative 2 over 3 end exponent plus e to the power of x equals fraction numerator 1 over denominator 3 cube root of x squared end root end fraction plus e to the power of x end style

(c) begin mathsize 20px style y equals x e to the power of x end style

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals x e to the power of x plus e to the power of x equals e to the power of x left parenthesis x plus 1 right parenthesis end style


مشتقة الاقتران الأسي الطبيعي، وقاعدة السلسة

أتحقق من فهمي صفحة 75

أجد مشتقة كل اقتران ممّا يأتي:

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

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 7 e to the power of 7 x plus 1 end exponent end style

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

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

(c) begin mathsize 20px style f left parenthesis x right parenthesis equals 5 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 fraction numerator 5 over denominator 2 square root of x end fraction e to the power of square root of x end exponent end style

 

أتحقق من فهمي صفحة 76

قمر صناعي: تُستعمل مادة مشعّة لتزويد قمر صناعي بالطاقة. ويمكن نمذجة مقدار الطاقة المتبقيّة في المادة المشعة (بالواط) باستعمال الاقتران: begin mathsize 20px style P left parenthesis t right parenthesis equals 50 e to the power of negative 0.004 t end exponent end style ، حيث t الزمن بالأيام. أجد معدل تغير الطاقة المتبقية في القمر الصناعي بعد 500 يوم، مفسراً معنى الناتج.

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تتناقص الطاقة المتبقية بمعدل 0.03 واط لكل يوم بعد 500 يوم.


مشتقة الاقتران اللوغاريتمي الطبيعي

أتحقق من فهمي صفحة 78

أجد مشتقة كل اقتران ممّا يأتي:

(a) begin mathsize 20px style f left parenthesis x right parenthesis equals 4 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 4 over x end style

(b) 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table end style

(c) 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table end style


مشتقة الاقتران اللوغاريتمي الطبيعي، وقاعدة السلسلة

أتحقق من فهمي صفحة 80

أجد مشتقة كل اقتران ممّا يأتي:

(a) 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table end style

(b) 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table end style

(c) 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table 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 left parenthesis x right parenthesis end cell cell equals left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of 1 third end exponent end cell row cell f to the power of straight prime left parenthesis x right parenthesis end cell cell equals 1 third left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis right parenthesis end cell row blank cell equals fraction numerator 2 plus 4 left parenthesis x squared plus x right parenthesis cubed left parenthesis 2 x plus 1 right parenthesis over denominator 3 cube root of left parenthesis 2 x plus left parenthesis x squared plus x right parenthesis to the power of 4 right parenthesis squared end root end fraction end cell end table end style

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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