Soal dan Pembahasan - Definisi Turunan Fungsi
Turunan fungsi adalah materi yang dipelajari pada jenjang SMA, tepatnya pada kelas XI. Turunan fungsi memiliki banyak penerapan, baik itu dalam bidang matematika maupun bidang lain, seperti fisika. Beberapa di antaranya adalah digunakan dalam menentukan gradien garis singgung kurva dan kecepatan sesaat dari suatu benda.
Oleh karena itu, penting bagi kita untuk mempelajari turunan fungsi. Mari kita mulai dengan definisi atau pengertian turunan fungsi.
Definisi
Misalkan $f$ adalah sebuah fungsi. Maka turunan dari $f$ adalah $f'$ (dibaca "f aksen"), dengan $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
Mari berlatih menentukan turunan fungsi, dengan menggunakan definisi turunan.
Soal dan Pembahasan
Tentukan turunan dari fungsi $f(x)=x$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h})=\textcolor{maroon}{x+h}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{x+h})-\textcolor{blue}{x}}{h} \\ &= \lim_{h \to 0} \frac{h}{h} \\ &= \lim_{h \to 0} 1 \\ &= 1 \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=2x-1$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = 2(\textcolor{maroon}{x+h})-1=2x+2h-1$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{2x+2h-1})-(\textcolor{blue}{2x-1})}{h} \\ &= \lim_{h \to 0} \frac{2h}{h} \\ &= \lim_{h \to 0} 2 \\ &= 2 \end{aligned}$$
Misalkan $f(x)=ax+b$, dengan $a,b \in \mathbb{R}$. Tentukan turunan dari fungsi $f(x)$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = a(\textcolor{maroon}{x+h})+b=ax+ah+b$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{ax+ah+b})-(\textcolor{blue}{ax+b})}{h} \\ &= \lim_{h \to 0} \frac{ah}{h} \\ &= \lim_{h \to 0} a \\ &= a \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=x^2$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = (\textcolor{maroon}{x+h})^2=x^2+2hx+h^2$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{x^2+2hx+h^2})-\textcolor{blue}{x^2}}{h} \\ &= \lim_{h \to 0} \frac{2hx+h^2}{h} \\ &= \lim_{h \to 0} \frac{h(2x+h)}{h} \\ &= \lim_{h \to 0} (2x+h) \\ &= 2x \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=3x^2+4$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= 3(\textcolor{maroon}{x+h})^2+4 \\ &= 3(x^2+2hx+h^2)+4 \\ &= 3x^2+6hx+3h^2+4 \end{aligned}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{3x^2+6hx+3h^2+4})-(\textcolor{blue}{3x^2+4})}{h} \\ &= \lim_{h \to 0} \frac{6hx+3h^2}{h} \\ &= \lim_{h \to 0} \frac{h(6x+3h)}{h} \\ &= \lim_{h \to 0} (6x+3h) \\ &= 6x \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=x^2+x+1$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= (\textcolor{maroon}{x+h})^2+(\textcolor{maroon}{x+h})+1 \\ &= (x^2+2hx+h^2)+(x+h)+1 \\ &= x^2+2hx+h^2+x+h+1 \end{aligned}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{x^2+2hx+h^2+x+h+1})-(\textcolor{blue}{x^2+x+1})}{h} \\ &= \lim_{h \to 0} \frac{2hx+h^2+h}{h} \\ &= \lim_{h \to 0} \frac{h(2x+h+1)}{h} \\ &= \lim_{h \to 0} (2x+h+1) \\ &= 2x+1 \end{aligned}$$
Misalkan $f(x)=ax^2+bx+c$, dengan $a,b,c \in \mathbb{R}$. Tentukan turunan dari fungsi $f(x)$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= a(\textcolor{maroon}{x+h})^2+b(\textcolor{maroon}{x+h})+c \\ &= a(x^2+2hx+h^2)+b(x+h)+c \\ &= ax^2+2ahx+ah^2+bx+bh+c \end{aligned}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{ax^2+2ahx+ah^2+bx+bh+c})-(\textcolor{blue}{ax^2+bx+c})}{h} \\ &= \lim_{h \to 0} \frac{2ahx+ah^2+bh}{h} \\ &= \lim_{h \to 0} \frac{h(2ax+ah+b)}{h} \\ &= \lim_{h \to 0} (2ax+ah+b) \\ &= 2ax+b \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=2x^3$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= 2(\textcolor{maroon}{x+h})^3 \\ &= 2(x^3+3hx^2+3h^2x+h^3) \\ &= 2x^3+6hx^2+6h^2x+2h^3 \end{aligned}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{2x^3+6hx^2+6h^2x+2h^3})-\textcolor{blue}{2x^3}}{h} \\ &= \lim_{h \to 0} \frac{6hx^2+6h^2x+2h^3}{h} \\ &= \lim_{h \to 0} \frac{h(6x^2+6hx+2h^2)}{h} \\ &= \lim_{h \to 0} (6x^2+6hx+2h^2) \\ &= 6x^2 \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=x^4$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = (\textcolor{maroon}{x+h})^4 = x^4+4x^3h+6x^2h^2+4xh^3+h^4$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{x^4+4x^3h+6x^2h^2+4xh^3+h^4})-\textcolor{blue}{x^4}}{h} \\ &= \lim_{h \to 0} \frac{4x^3h+6x^2h^2+4xh^3+h^4}{h} \\ &= \lim_{h \to 0} \frac{h(4x^3+6x^2h+4xh^2+h^3)}{h} \\ &= \lim_{h \to 0} (4x^3+6x^2h+4xh^2+h^3) \\ &= 4x^3 \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=x^3+2x^2+1$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= (\textcolor{maroon}{x+h})^3+2(\textcolor{maroon}{x+h})^2+1 \\ &= (x^3+3hx^2+3h^2x+h^3)+2(x^2+2hx+h^2)+1 \\ &= x^3+3hx^2+3h^2x+h^3+2x^2+4hx+2h^2+1 \end{aligned}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{(\textcolor{green}{x^3+3hx^2+3h^2x+h^3+2x^2+4hx+2h^2+1})-(\textcolor{blue}{x^3+2x^2+1})}{h} \\ &= \lim_{h \to 0} \frac{3hx^2+3h^2x+h^3+4hx+2h^2}{h} \\ &= \lim_{h \to 0} \frac{h(3x^2+3hx+h^2+4x+2h)}{h} \\ &= \lim_{h \to 0} (3x^2+3hx+h^2+4x+2h) \\ &= 3x^2+4x \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=x^4+x^2$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= (\textcolor{maroon}{x+h})^4+(\textcolor{maroon}{x+h})^2 \\ &= (x^4+4hx^3+6h^2x^2+4h^3x+h^4)+(x^2+2hx+h^2) \end{aligned}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{\textcolor{green}{(x^4+4hx^3+6h^2x^2+4h^3x+h^4)+(x^2+2hx+h^2)}-(\textcolor{blue}{x^4+x^2})}{h} \\ &= \lim_{h \to 0} \frac{4hx^3+6h^2x^2+4h^3x+h^4+2hx+h^2}{h} \\ &= \lim_{h \to 0} \frac{h(4x^3+6hx^2+4h^2x+h^3+2x+h)}{h} \\ &= \lim_{h \to 0} (4x^3+6hx^2+4h^2x+h^3+2x+h) \\ &= 4x^3+2x \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=\dfrac{1}{x}$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = \frac{1}{\textcolor{maroon}{x+h}}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{\textcolor{green}{\dfrac{1}{x+h}}-\textcolor{blue}{\dfrac{1}{x}}}{h} \\ &= \lim_{h \to 0} \left( \frac{1}{x+h}-\frac{1}{x} \right) \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \frac{x-(x+h)}{(x+h)x} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \frac{-h}{(x+h)x} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} -\frac{1}{(x+h)x} \\ &= -\frac{1}{(x+0)x} \\ &= -\frac{1}{x^2} \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=\dfrac{1}{x+1}$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = \frac{1}{(\textcolor{maroon}{x+h})+1} = \frac{1}{x+h+1}$$ sehingga $$\begin{aligned} \textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)} &= \textcolor{green}{\dfrac{1}{x+h+1}}-\textcolor{blue}{\dfrac{1}{x+1}} \\ &= \frac{(x+1)-(x+h+1)}{(x+h+1)(x+1)} \\ &= \frac{-h}{(x+h+1)(x+1)} \end{aligned}$$
Berdasarkan definisi turunan fungsi diperoleh $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{maroon}{f(x+h)-f(x)}}{h} \\ &= \lim_{h \to 0} \frac{\textcolor{maroon}{\dfrac{-h}{(x+h+1)(x+1)}}}{h} \\ &= \lim_{h \to 0} \frac{-h}{(x+h+1)(x+1)} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} -\frac{1}{(x+h+1)(x+1)} \\ &= -\frac{1}{(x+0+1)(x+1)} \\ &= -\frac{1}{(x+1)^2} \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=\dfrac{2}{x^2}$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = \frac{2}{(\textcolor{maroon}{x+h})^2}$$ sehingga $$\begin{aligned} \textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)} &= \textcolor{green}{\frac{2}{(x+h)^2}}-\textcolor{blue}{\frac{2}{x^2}} \\ &= \frac{2x^2-2(x+h)^2}{(x+h)^2x^2} \\ &= \frac{2x^2-(2x^2+4hx+2h^2)}{(x+h)^2x^2} \\ &= \frac{-4hx-2h^2}{(x+h)^2x^2} \end{aligned}$$
Berdasarkan definisi turunan fungsi diperoleh $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{maroon}{f(x+h)-f(x)}}{h} \\ &= \lim_{h \to 0} (\textcolor{maroon}{f(x+h)-f(x)}) \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \textcolor{maroon}{\frac{-4hx-2h^2}{(x+h)^2x^2}} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \frac{h(-4x-2h)}{(x+h)^2x^2} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \frac{-4x-2h}{(x+h)^2x^2} \\ &= \frac{-4x-2 \cdot 0}{(x+0)^2x^2} \\ &= \frac{-4x}{x^2x^2} \\ &= -\frac{4}{x^2} \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=\dfrac{x-1}{x+1}$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = \frac{\textcolor{maroon}{x+h}-1}{\textcolor{maroon}{x+h}+1}$$ sehingga $$\begin{aligned} \textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)} &= \textcolor{green}{\frac{x+h-1}{x+h+1}}-\textcolor{blue}{\frac{x-1}{x+1}} \\ &= \frac{(x+h-1)(x+1)-(x+h+1)(x-1)}{(x+h+1)(x+1)} \\ &= \frac{(x^2+hx+h-1)-(x^2+hx-h-1)}{(x+h+1)(x+1)} \\ &= \frac{2h}{(x+h+1)(x+1)} \end{aligned}$$
Berdasarkan definisi turunan fungsi diperoleh $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{maroon}{f(x+h)-f(x)}}{h} \\ &= \lim_{h \to 0} (\textcolor{maroon}{f(x+h)-f(x)}) \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \textcolor{maroon}{\frac{2h}{(x+h+1)(x+1)}} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \frac{2}{(x+h+1)(x+1)} \\ &= \frac{2}{(x+0+1)(x+1)} \\ &= \frac{2}{(x+1)^2} \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=\dfrac{2x-1}{x-4}$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = \frac{2(\textcolor{maroon}{x+h})-1}{\textcolor{maroon}{x+h}-4} = \frac{2x+2h-1}{x+h-4}$$ sehingga $$\begin{aligned} \textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)} &= \textcolor{green}{\frac{2x+2h-1}{x+h-4}}-\textcolor{blue}{\frac{2x-1}{x-4}} \\ &= \frac{(2x+2h-1)(x-4)-(x+h-4)(2x-1)}{(x+h-4)(x-4)} \\ &= \frac{(2x^2+2hx-9x-8h+4)-(2x^2+2hx-9x-h+4)}{(x+h-4)(x-4)} \\ &= \frac{-7h}{(x+h-4)(x-4)} \end{aligned}$$
Berdasarkan definisi turunan fungsi diperoleh $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{maroon}{f(x+h)-f(x)}}{h} \\ &= \lim_{h \to 0} (\textcolor{maroon}{f(x+h)-f(x)}) \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \textcolor{maroon}{\frac{-7h}{(x+h-4)(x-4)}} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} -\frac{7}{(x+h-4)(x-4)} \\ &= -\frac{7}{(x+0-4)(x-4)} \\ &= -\frac{7}{(x-4)^2} \end{aligned}$$
Tentukan turunan dari fungsi $f(x)=\dfrac{x}{x^2-x}$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$\begin{aligned} f(\textcolor{maroon}{x+h}) &= \frac{\textcolor{maroon}{x+h}}{(\textcolor{maroon}{x+h})^2-(\textcolor{maroon}{x+h})} \\ &= \frac{x+h}{(x^2+2hx+h^2)-(x+h)} \\ &= \frac{x+h}{x^2+2hx+h^2-x-h} \end{aligned}$$ sehingga $$\begin{aligned} \textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)} &= \textcolor{green}{\frac{x+h}{x^2+2hx+h^2-x-h}}-\textcolor{blue}{\frac{x}{x^2-x}} \\ &= \frac{(x+h)(x^2-x)-x(x^2+2hx+h^2-x-h)}{(x^2+2hx+h^2-x-h)(x^2-x)} \\ &= \frac{(x^3 - x^2 + hx^2 - hx)-(x^3 + 2hx^2 + h^2x - x^2 - hx)}{(x^2+2hx+h^2-x-h)(x^2-x)} \\ &= \frac{-hx^2-h^2x}{(x^2+2hx+h^2-x-h)(x^2-x)} \\ &= \frac{h(-x^2-hx)}{(x^2+2hx+h^2-x-h)(x^2-x)} \end{aligned}$$
Berdasarkan definisi turunan fungsi diperoleh $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{maroon}{f(x+h)-f(x)}}{h} \\ &= \lim_{h \to 0} (\textcolor{maroon}{f(x+h)-f(x)}) \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \textcolor{maroon}{\frac{h(-x^2-hx)}{(x^2+2hx+h^2-x-h)(x^2-x)}} \cdot \frac{1}{h} \\ &= \lim_{h \to 0} \frac{-x^2-hx}{(x^2+2hx+h^2-x-h)(x^2-x)} \\ &= \frac{-x^2}{(x^2-x)(x^2-x)} \\ &= \frac{-1}{(x-1)(x-1)} \\ &= -\frac{1}{(x-1)^2} \end{aligned}$$
Misalkan $f(x)=\sqrt{x}$, dengan $x > 0$. Tentukan turunan dari fungsi $f(x)$ di titik $x$.
Berdasarkan definisi fungsi $f$, diperoleh $$f(\textcolor{maroon}{x+h}) = \sqrt{\textcolor{maroon}{x+h}}$$ sehingga $$\begin{aligned} f'(x) &= \lim_{h \to 0} \frac{\textcolor{green}{f(x+h)}-\textcolor{blue}{f(x)}}{h} \\ &= \lim_{h \to 0} \frac{\textcolor{green}{\sqrt{x+h}}-\textcolor{blue}{\sqrt{x}}}{h} \\ &= \lim_{h \to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \\ &= \lim_{h \to 0} \frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})} \\ &= \lim_{h \to 0} \frac{h}{h(\sqrt{x+h}+\sqrt{x})} \\ &= \lim_{h \to 0} \frac{1}{\sqrt{x+h}+\sqrt{x}} \\ &= \frac{1}{\sqrt{x}+\sqrt{x}} \\ &= \frac{1}{2\sqrt{x}} \end{aligned}$$