كتاب التمارين

المشتقة الثانية والسرعة المتجهة والتسارع

أجد المشتقة الثانية لكل اقتران مما يأتي:

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

$\begin{array}{rl}& f^{\mathrm{\prime }}\left(x\right)=15x^2+4\\ & f^{\prime \prime }\left(x\right)=30x\end{array}$

(2) $f\left(x\right)=5e^{4x}$

$\begin{array}{rl}& f^{\mathrm{\prime }}\left(x\right)=20e^{4x}\\ & f^{\prime \prime }\left(x\right)=80e^{4x}\end{array}$

(3) $f\left(x\right)=\sqrt[3]{x}$

$\begin{array}{rl}& f\left(x\right)=\sqrt[3]{x}=x^{\frac{1}{3}}\\ & f^{\mathrm{\prime }}\left(x\right)=\frac{1}{3}{x}^{-\frac{2}{3}}\\ & f^{\prime \prime }\left(x\right)=-\frac{2}{9}{x}^{-\frac{5}{3}}=-\frac{2}{9\sqrt[3]{x^5}}\end{array}$

(4) $f\left(x\right)=7\ln x$

$\begin{array}{rl}& f^{\mathrm{\prime }}\left(x\right)=\frac{7}{x}\\ & f^{\prime \prime }\left(x\right)=\frac{-7}{x^2}\end{array}$

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

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

(6) $f\left(x\right)=e^x\sin x$

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =\left(e^x\right)(cos x)+(sin x)\left(e^x\right)\\ & =e^x(cos x+sin x)\\ f^{\prime \prime }\left(x\right)& =e^x(-sin x+cos x)+e^x(cos x+sin x)\\ & =-e^{x}sin x+e^{x}cos x+e^x cos x+e^x sin x\\ & =2e^{x}cos x\end{array}$