إجابات كتاب التمارين

مشتقتا الضرب والقسمة

أجد مشتقة كل اقتران ممّا يأتي:

(1) $f\left(x\right)=2x{(1+3x^2)}^3$

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =\left(2x\right)\times 3{(1+3x^2)}^2\left(6x\right)+{(1+3x^2)}^3\left(2\right)\\ & =36x^2{(1+3x^2)}^2+2{(1+3x^2)}^3\\ & ={(1+3x^2)}^2(42x^2+2)\end{array}$

(2) $f\left(x\right)=\frac{x-2}{x+2}$

$f^{\mathrm{\prime }}\left(x\right)=\frac{(x+2)\left(1\right)-(x-2)\left(1\right)}{(x+2{)}^2}=\frac{4}{(x+2{)}^2}$

(3) $f\left(x\right)=\frac{x^3-1}{x^2+1}+4x^3$

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =\frac{(x^2+1)\left(3x^2\right)-(x^3-1)\left(2x\right)}{{(x^2+1)}^2}+12x^2\\ & =\frac{3x^4+3x^2-2x^4+2x}{{(x^2+1)}^2}+12x^2\\ & =\frac{x^4+3x^2+2x}{{(x^2+1)}^2}+12x^2\end{array}$

(4) $f\left(x\right)={(1-x^2)}^4(2x+6{)}^3$

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& ={(1-x^2)}^4\times 3(2x+6{)}^2(2)+(2x+6{)}^3\times 4{(1-x^2)}^3(-2x)\\ & =6{(1-x^2)}^4(2x+6{)}^2-8x(2x+6{)}^3{(1-x^2)}^3\\ & =2{(1-x^2)}^3(2x+6{)}^2(3-24x-11x^2)\end{array}$

(5) $f\left(x\right)=\frac{3x+5}{(x+1{)}^2}$

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =\frac{(x+1{)}^2(3)-(3x+5\left)\right(2\left)\right(x+1)}{(x+1{)}^4}\\ & =\frac{-3x^2-10x-7}{(x+1{)}^4}\end{array}$

(6) $f\left(x\right)=(5x^2+4x-3)(2x^2-3x+1)$

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =(5x^2+4x-3)(4x-3)+(2x^2-3x+1)(10x+4)\\ & =40x^3-21x^2-26x+13\end{array}$

(7) $f\left(x\right)=(3x^5-x^2)(x-\frac{5}{x})$

بفك الأقواس:

$\begin{array}{rl}& f\left(x\right)=3x^6-15x^4-x^3+5x\\ & f^{\mathrm{\prime }}\left(x\right)=18x^5-60x^3-3x^2+5\end{array}$

أو بتطبيق قاعدة مشتقة ضرب اقترانين:

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =(3x^5-x^2)(1+\frac{5}{x^2})+(x-\frac{5}{x})(15x^4-2x)\\ & =18x^5-60x^3-3x^2+5\end{array}$

(8) $f\left(x\right)=\frac{5x^2-1}{2x^3+3}$

$f^{\mathrm{\prime }}\left(x\right)=\frac{(2x^3+3)\left(10x\right)-(5x^2-1)\left(6x^2\right)}{{(2x^3+3)}^2}=\frac{-10x^4+6x^2+30x}{{(2x^3+3)}^2}$

(9) $f\left(x\right)=\frac{1}{x-4}$

$f^{\mathrm{\prime }}\left(x\right)=\frac{-1}{(x-4{)}^2}$