Contoh Soal 1 Turunan Fungsi Aljabar

 $\begin{array}{l}\\ 1.& \text{Diketahui}f{}^{\prime }(2)=\underset{x\to 2}{\text{Lim}}{\displaystyle \frac{x^3-8}{x-2},}\\ & \text{maka fungsi}f(x)=....\\ & \begin{array}{l}\\ \text{a}.12& & \text{d}.x^3\\ \text{b}.2& \text{c}.x& \text{e}.x-8\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \text{Turunan fungsi f di}x=c\text{adalah}\\ & f{}^{\prime }(c)=\underset{x\to c}{\text{Lim}}{\displaystyle \frac{f(x)-f(c)}{x-c},\text{maka}}\\ & \text{turunan fungsi f di}x=2\text{adalah}\\ & f{}^{\prime }(2)=\underset{x\to 2}{\text{Lim}}{\displaystyle \frac{x^3-8}{x-2}}\\ & \text{sehingga akan didapa}\text{tkan fungsi}f\text{nya }\\ & \text{yaitu}f(x)=x^3\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Jika}a\ne 0,\text{maka nilai dari}\\ & \underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \sqrt[3]{x}-{\displaystyle \sqrt[3]{a}}}}{x-a}=....}\\ & \begin{array}{l}\\ \text{a}.3a{\displaystyle \sqrt[3]{a}}& & \text{d}.{\displaystyle \frac{1}{2a}\sqrt[3]{a}}\\ \text{b}.2a{\displaystyle \sqrt[3]{a}}& \text{c}.0& \text{e}.{\displaystyle \frac{1}{3a}\sqrt[3]{a}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Alternatif 1}\\ & \begin{array}{rl}f{}^{\prime }(a)& =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \sqrt[3]{x}-{\displaystyle \sqrt[3]{a}}}}{x-a}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \sqrt[3]{x}-{\displaystyle \sqrt[3]{a}}}}{(\sqrt[3]{x}-\sqrt[3]{a})(\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2})}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{1}{(\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2})}}\\ & ={\displaystyle \frac{1}{(\sqrt[3]{a^2}+\sqrt[3]{a.a}+\sqrt[3]{a^2})}}\\ & ={\displaystyle \frac{1}{3\sqrt[3]{a^2}}}\\ & ={\displaystyle \frac{1}{3\sqrt[3]{a^2}}\times {\displaystyle \frac{\sqrt[3]{a}}{\sqrt[3]{a}}}}\\ & ={\displaystyle \frac{1}{3a}\sqrt[3]{a}}\end{array}\\ & \text{Alternatif 2 (dengan aturan L'Hopital)}\\ & \begin{array}{rl}\underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \sqrt[3]{x}-{\displaystyle \sqrt[3]{a}}}}{x-a}}& =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{f(x)}{g(x)}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{f{}^{\prime }(x)}{g{}^{\prime }(x)}}\\ \underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \sqrt[3]{x}-{\displaystyle \sqrt[3]{a}}}}{x-a}}& =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{x^{{}^{\frac{1}{3}}}-a^{{}^{\frac{1}{3}}}}{x-a}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \frac{1}{3}{x}^{{}^{\frac{1}{3}-1}}-0}}{1-0}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{{\displaystyle \frac{1}{3}{x}^{{}^{-\frac{2}{3}}}}}{1}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{1}{3\sqrt[3]{x^2}}\times \frac{\sqrt[3]{x}}{\sqrt[3]{x}}}\\ & =\underset{x\to a}{\text{Lim}}{\displaystyle \frac{1}{3x}\sqrt[3]{x}}\\ & ={\displaystyle \frac{1}{3a}\sqrt[3]{a}}\end{array}\end{array}$ .

$\begin{array}{l}\\ 3.& \text{Jika}f(x)={\displaystyle \frac{1}{\sqrt{x}}\text{maka nilai dari}}\\ & -2f{}^{\prime }(x)=....\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{1}{x\sqrt{x}}}& & \text{d}.{\displaystyle -\frac{1}{2x\sqrt{x}}}\\ \text{b}.{\displaystyle x\sqrt{x}}& \text{c}.{\displaystyle -\frac{1}{2\sqrt{x}}}& \text{e}.{\displaystyle -2x\sqrt{x}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{Dike}& \text{tahui}\\ f(x)& ={\displaystyle \frac{1}{\sqrt{x}}={\displaystyle \frac{1}{x^{{}^{\frac{1}{2}}}}=x^{{}^{-\frac{1}{2}}}}}\end{array}\\ & \begin{array}{|cc|}\hline y=a{x}^n\to y{}^{\prime }=na{x}^{n-1}& \begin{array}{rl}& \\ y& ={\displaystyle \frac{U}{V}\to y{}^{\prime }={\displaystyle \frac{U^{\prime }.V-U.V^{\prime }}{V^2}}}\\ & \end{array}\\ \begin{array}{rl}f{}^{\prime }(x)& =-{\displaystyle \frac{1}{2}{x}^{{}^{-\frac{1}{2}-1}}}\\ & =-{\displaystyle \frac{1}{2}{x}^{{}^{-\frac{3}{2}}}}\\ & =-{\displaystyle \frac{1}{2x^{{}^{\frac{3}{2}}}}}\\ & =-{\displaystyle \frac{1}{2x^1.x^{{}^{\frac{1}{2}}}}}\\ & =-{\displaystyle \frac{1}{2x\sqrt{x}}}\end{array}& \begin{array}{rl}f{}^{\prime }(x)& ={\displaystyle \frac{0.\sqrt{x}-1.\frac{1}{2}{x}^{{}^{\frac{1}{2}-1}}}{{\left(\sqrt{x}\right)}^2}}\\ & ={\displaystyle \frac{-{\displaystyle \frac{1}{2}{x}^{{}^{-\frac{1}{2}}}}}{x}}\\ & =-{\displaystyle \frac{1}{2x{x}^{{}^{\frac{1}{2}}}}}\\ & =-{\displaystyle \frac{1}{2x\sqrt{x}}}\end{array}\\ \hline\end{array}\end{array}$.

$\begin{array}{l}\\ 4.& \text{Turunan pertama dari}y={\displaystyle \sqrt[n]{x}\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.{\displaystyle \frac{1}{n}{x}^{{}^{\frac{1}{n}}}}& & \text{d}.(n-1)\sqrt[n-1]{x}\\ \text{b}.{\displaystyle \frac{1}{n}{x}^{{}^{\frac{1-n}{n}}}}& \text{c}.{\displaystyle \frac{1}{n-1}{x}^{{}^{n-1}}}& \text{e}.\sqrt[n-1]{x}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}y& =\sqrt[n]{x}={\displaystyle x^{{}^{\frac{1}{n}}},\text{maka}}\\ y{}^{\prime }& ={\displaystyle \frac{1}{n}{x}^{{}^{\frac{1}{n}-1}}={\displaystyle \frac{1}{n}{x}^{{}^{\frac{1-n}{n}}}}}\end{array}\end{array}$.

$\begin{array}{l}\\ 5.& \text{Turunan ke}-n\text{dari}y={\displaystyle \frac{1}{x}\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.{\displaystyle n!.x^{-(n+1)}}& \\ \text{b}.{\displaystyle (n+1)!.x^{-(n+1)}}& \\ \text{c}.{\displaystyle (-1{)}^{n}n!.x^{-(n+1)}}\\ \text{d}.(-1{)}^{n+1}n!.x^{-(n+1)}\\ \text{e}.(-1{)}^{n+1}(n+1)!.x^{-(n+1)}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{|ccl|}\hline \text{Fungsi}& y={\displaystyle \frac{1}{x}}& y=x^{-1}\\ y^{\prime }={\displaystyle \frac{dy}{dx}}& -x^{-2}& (-1{)}^1.1!.x^{-(1+1)}\\ y^{″}={\displaystyle \frac{d^{2}y}{d{x}^2}}& 2x^{-3}& (-1{)}^2.2!.x^{-(2+1)}\\ y^{‴}={\displaystyle \frac{d^{3}y}{d{x}^3}}& -6x^{-4}& (-1{)}^3.3!.x^{-(3+1)}\\ y^{IV}={\displaystyle \frac{d^{4}y}{d{x}^4}}& 24x^{-5}& (-1{)}^4.4!.x^{-(4+1)}\\ y^V={\displaystyle \frac{d^{5}y}{d{x}^5}}& \cdots & \cdots \\ ⋮& ⋮& ⋮\\ y^n={\displaystyle \frac{d^{n}y}{d{x}^n}}& \cdots & (-1{)}^n.n!.x^{-(n+1)}\\ \hline\end{array}\end{array}$.