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أتحقق من فهمي

مشتقتا الضرب والقسمة والمشتقات العليا

أتحقق من فهمي صفحة 30

مشتقة ضرب اقترانين

أجد مشتقة كل اقتران ممّا يأتي:

(a) f(x) = (x3 – 2x2 + 3) (7x2 – 4x)

f ’(x) = (x3 – 2x2 + 3) (14x – 4) + (7x2 – 4x) (3x2 – 4x)

f ’(x) = 14x4 – 4x3 – 28x3 + 8x2 + 42x – 12 + 21x4 – 12x3 + 16x2

f ’(x) = 35x4 – 72x3 + 24x2 + 42x - 12

(b) f(x) = ln x cos x

f ’(x) = (ln x) (-sin x) + (cos x) (begin mathsize 20px style 1 over x end style) = - ln x sin x + begin mathsize 20px style fraction numerator cos space x over denominator x end fraction end style

أتحقق من فهمي صفحة 32

مشتقة قسمة اقترانين

أجد مشتقة كل اقتران ممّا يأتي:

(a) f(x) = begin mathsize 20px style fraction numerator x space plus space 1 over denominator 2 x space plus space 1 end fraction end style

f ’(x) = begin mathsize 20px style fraction numerator left parenthesis 2 x space plus space 1 right parenthesis space left parenthesis 1 right parenthesis space minus space left parenthesis x space plus space 1 right parenthesis space left parenthesis 2 right parenthesis over denominator left parenthesis 2 x space plus space 1 right parenthesis squared end fraction end style = begin mathsize 20px style fraction numerator negative 1 over denominator left parenthesis 2 x space plus space 1 right parenthesis squared end fraction end style

(b) f(x) = begin mathsize 20px style fraction numerator sin space x over denominator e to the power of x end fraction end style

f ’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style = begin mathsize 20px style B equals fraction numerator mu subscript straight o I over denominator 2 pi r end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 50 over denominator 2 pi cross times 2.5 end fraction equals 4 cross times 10 to the power of negative 6 end exponent straight T end style

أتحقق من فهمي صفحة 34

مشتقة قسمة اقترانين

سكان: يعطى عدد سكان مدينة صغيرة بالاقتران: begin mathsize 20px style B equals fraction numerator mu subscript straight o I over denominator 2 pi r end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 50 over denominator 2 pi cross times 2.5 end fraction equals 4 cross times 10 to the power of negative 6 end exponent straight T end style P(t) =، حيث t الزمن بالسنوات، و P عدد السكان بالآلاف:

(a) أجد معدل تغير عدد السكان في المدينة بالنسبة إلى الزمن.

P’(t) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

(b) أجد معدل تغير عدد السكان في المدينة عندما t = 12 مفسراً معنى الناتج.

P’(12) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style ≈ 231.405

إذن في السنة 12 يتزايد عدد سكان هذه المدينة بمعدل 231 ألف نسمة سنوياً تقريباً.

أتحقق من فهمي صفحة 35

مشتقة المقلوب

أجد مشتقة كل اقتران ممّا يأتي:

(a) f(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

f ’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

(b) f(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

f ’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style = - begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

أتحقق من فهمي صفحة 37

مشتقة الاقترانات المثلثية

أجد مشتقة كل اقتران ممّا يأتي:

(a) f(x) = x cot x

f ’(x) = (x) (-csc2 x) + (cot x) (1) = -x csc2 x + cot x

(b) f(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

f ’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style 

f ’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

أتحقق من فهمي صفحة 38

المشتقة العليا

أجد المشتقات الثلاث الأولى للاقتران: begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style f(x) =

f ’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style  = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style - begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

f ’’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style - begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

f ’’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style - begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style

f ’’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style 

f ’’’(x) = begin mathsize 20px style B equals fraction numerator mu subscript straight o I N over denominator 2 R end fraction equals fraction numerator 4 pi cross times 10 to the power of negative 7 end exponent cross times 0.4 cross times 100 over denominator 2 cross times 0.08 end fraction equals 3.14 cross times 10 to the power of negative 4 end exponent straight T end style 

ويمكن التوصل إلى الإجابة نفسها بتحويل الاقتران إلى f(x) = x-1 sin x وتطبيق قاعدة مشتقة ضرب اقترانين.

إعداد : شبكة منهاجي التعليمية

20 / 09 / 2022

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