Tanya Jawab 1 : Turunan

Berikut ini adalah jawaban dari pertanyaan-pertanyaan yang masuk

Semoga membantu….

Tanya :

Tentukan turunan dari $\large f(x)=log\;2\left (\frac{sin\;x^2+e^{x^2}}{x+1} \right )$

Jawab :

  • turunan fungsi logaritma :

    ${\color{Red} f(x)=^alog\;g(x)}\;\;\;\;maka\;\;\;\;\;{\color{Red} f'(x)=\frac{g'(x)}{g(x)}\;^alog\;e}$

  • $f(x)=log\;2\left (\frac{sin\;x^2+e^{x^2}}{x+1} \right )$ adalah fungsi logaritma dengan bilangan basis $a = 10$ , dan $g(x)= 2\left (\frac{sin\;x^2+e^{x^2}}{x+1} \right )$

  • kita cari turunan dari fungsi $g(x)$ terlebih dahulu :

    \begin{align*}g(x) & = & 2\left (\frac{sin\;x^2+e^{x^2}}{x+1} \right )\\ & = & \frac{2sin\;x^2+2e^{x^2}}{x+1}\end{align*}

    ingat turunan fungsi pembagian !! kita misalkan :

    \begin{array}{lcl}u=2sin\;x^2+2e^{x^2} & maka & u'=4x.cos\;x^2+4x.e^{x^2}\\v=x+1 & maka & v'=1\end{array}

    maka :

    \begin{align*}g'(x) & = & \frac{u'.v-v'.u}{v^2}\\ & = & \frac{4x.cos\;x^2+4x.e^{x^2}(x+1)-1.(2.sin\;x^2+2.e^{x^2})}{(x+1)^2}\\ & = & \frac{4x^2.cos\;x^2+4x^2.e^{x^2}+4x.cos\;x^2+4x.e^{x^2}-2.sin\;x^2-2.e^{x^2}}{(x+1)^2}\\ & = & \frac{2(2x^2.cos\;x^2+2x^2.e^{x^2}+2x.cos\;x^2+2x.e^{x^2}-sin\;x^2-e^{x^2})}{(x+1)^2}\end{align*}

    kembali ke fungsi $f(x)$ , maka kita akan dapatkan turunan :

    \begin{align*}f'(x) & = & \frac{g'(x)}{g(x)}\;^alog\;e\\ & = & \left (\frac{\frac{2(2x^2.cos\;x^2+2x^2.e^{x^2}+2x.cos\;x^2+2x.e^{x^2}-sin\;x^2-e^{x^2})}{(x+1)^2}}{\frac{2(sin\;x^2+e^{x^2})}{(x+1)}} \right )log\;e\\ & = & \left (\frac{2x^2.cosx^2+2x^2.e^{x^2}+2x.cosx^2+2x.e^{x^2}-sinx^2-e^{x^2}}{(x+1)(sinx^2+e^{x^2})} \right )log\;e\\ & = & \left (\frac{2x^2.cosx^2+2x^2.e^{x^2}+2x.cosx^2+2x.e^{x^2}-sinx^2-e^{x^2}}{x.sinx^2+sinx^2+x.e^{x^2}+e^{x^2}} \right )log\;e\end{align*}

    Nah…seperti inilah hasil turunannya :

    \begin{align*}f'(x) & = & \frac{g'(x)}{g(x)}\;^alog\;e\\ & = & \left (\frac{\frac{2(2x^2.cos\;x^2+2x^2.e^{x^2}+2x.cos\;x^2+2x.e^{x^2}-sin\;x^2-e^{x^2})}{(x+1)^2}}{\frac{2(sin\;x^2+e^{x^2})}{(x+1)}} \right )log\;e\\ & = & \left (\frac{2x^2.cosx^2+2x^2.e^{x^2}+2x.cosx^2+2x.e^{x^2}-sinx^2-e^{x^2}}{(x+1)(sinx^2+e^{x^2})} \right )log\;e\\ & = & \left (\frac{2x^2.cosx^2+2x^2.e^{x^2}+2x.cosx^2+2x.e^{x^2}-sinx^2-e^{x^2}}{x.sinx^2+sinx^2+x.e^{x^2}+e^{x^2}} \right )log\;e\end{align*}

Comments