Contoh Soal 1 Turunan Fungsi Trigonometri (Bagian 1)

$\begin{array}{l}\\ 1.& \text{Diketahui}f(x)=2\cos x-2020\\ & \text{Turunan pertama fungsi}f(x)\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 2\sin x\\ \text{b}.& -2\sin x\\ \text{c}.& -2\sin x-2020x\\ \text{d}.& 2{\sin }^{2}x\\ \text{e}.& 2\cos x-2020x\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}f(x)& =2\cos x-2020\\ f^{\prime }(x)& =-2\sin x\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Jika}{f}^{\prime }(x)\text{adalah turunan pertama dari}\\ & \text{fungsi}f(x)={\sin }^{7}x,\text{maka}{f}^{\prime }(x)=....\\ & \begin{array}{l}\\ \text{a}.& 7{\cos }^{6}x\\ \text{b}.& 7{\cos }^{7}x\\ \text{c}.& 7{\sin }^{6}x\cos x\\ \text{d}.& 7{\cos }^{6}x\sin x\\ \text{e}.& 7{\cos }^{6}x{\sin }^{6}x\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}f(x)& ={\sin }^{7}x\\ \text{guna}& \text{kan formula}:y=a.u^n\Rightarrow y^{\prime }=n.a.u^{n-1}.u^{\prime }\\ f^{\prime }(x)& =7{\sin }^{6}x(\cos x)=7{\sin }^{6}x\cos x\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Turunan pertama fungsi}g(x)=-5{\sin }^{3}x\\ & \text{adalah}{g}^{\prime }(x)=....\\ & \begin{array}{l}\\ \text{a}.& -5{\sin }^{2}x\cos x\\ \text{b}.& -5{\sin }^2{\cos }^{2}x\\ \text{c}.& -15{\sin }^{2}x\cos x\\ \text{d}.& -15{\cos }^{3}x\\ \text{e}.& -15{\sin }^{4}x\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}g(x)& =-5{\sin }^{3}x\\ \text{guna}& \text{kan formula}:y=a.u^n\Rightarrow y^{\prime }=n.a.u^{n-1}.u^{\prime }\\ g^{\prime }(x)& =-5(3{\sin }^{2}x)(\cos x)=-15{\sin }^{2}x\cos x\end{array}\end{array}$

$\begin{array}{l}\\ 4.& \text{Jika}h(x)=4x^3+\sin x+\cos x\\ & \text{maka}{h}^{\prime }(x)=....\\ & \begin{array}{l}\\ \text{a}.& 12x^2+\cos x-\sin x\\ \text{b}.& 12x^2-\cos x+\sin x\\ \text{c}.& 4x^3-\cos x-\sin x\\ \text{d}.& 4x^3-\sin x-\cos x\\ \text{e}.& 12x^3+\cos x+\sin x\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}h(x)& =4x^3+\sin x+\cos x\\ \text{guna}& \text{kan formula}:y=a.u^n\Rightarrow y^{\prime }=n.a.u^{n-1}.u^{\prime }\\ \text{pada}& \text{fungsi aljabarnya, yaitu}:y=4x^3\Rightarrow y^{\prime }=12x^2\\ \text{seda}& \text{ngkan fungsi transendennya mengikuti}\\ \text{turu}& \text{nan fungsi trigonometri biasa. Sehingga}\\ f^{\prime }(x)& =12x^2+\cos x-\sin x\end{array}\end{array}$

$\begin{array}{l}\\ 5.& \text{Jika}p(x)=-{\cos }^{4}x,\text{maka nilai}\\ & \text{maka}{p}^{\prime }\left({\displaystyle \frac{\pi }{3}}\right)=....\\ & \begin{array}{l}\\ \text{a}.& 0\\ \text{b}.& \sqrt{3}\\ \text{c}.& {\displaystyle \frac{1}{2}\sqrt{3}}\\ \text{d}.& {\displaystyle \frac{1}{4}\sqrt{3}}\\ \text{e}.& 1\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}p(x)& =-{\cos }^{4}x\\ p^{\prime }(x)& =-4{\cos }^{3}x.(-\sin x)=4{\cos }^{3}x\sin x\\ p^{\prime }\left({\displaystyle \frac{\pi }{3}}\right)& =4{\cos }^3\left({\displaystyle \frac{\pi }{3}}\right).\sin \left({\displaystyle \frac{\pi }{3}}\right)\\ & =4{\cos }^3{60}^{\circ }\times \sin {60}^{\circ }\\ & =4{\left({\displaystyle \frac{1}{2}}\right)}^3\times \left({\displaystyle \frac{1}{2}\sqrt{3}}\right)\\ & ={\displaystyle \frac{4}{16}\sqrt{3}}\\ & ={\displaystyle \frac{1}{4}\sqrt{3}}\end{array}\end{array}$