Whakaoti i te f^prime(x)=1/x-2ax+2-a

$$f ^ { \prime } ( x ) = \frac { 1 } { x } - 2 a x + 2 - a$$

Whakaoti mō a (complex solution)

$$\left\{\begin{matrix}a=\frac{1}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&x=-\frac{1}{2}\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xx=1-2axx+x\times 2-ax$$

Whakareatia te x ki te x, ka x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}=1-2axx+x\times 2-ax$$

Whakareatia te x ki te x, ka x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}=1-2ax^{2}+x\times 2-ax$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1-2ax^{2}+x\times 2-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}$$

Tangohia te 1 mai i ngā taha e rua.

$$-2ax^{2}+x\times 2-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1$$

Tangohia te x\times 2 mai i ngā taha e rua.

$$-2ax^{2}-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1-x\times 2$$

Whakareatia te -1 ki te 2, ka -2.

$$-2ax^{2}-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1-2x$$

Pahekotia ngā kīanga tau katoa e whai ana i te a.

$$\left(-2x^{2}-x\right)a=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1-2x$$

He hanga arowhānui tō te whārite.

$$\left(-2x^{2}-x\right)a=-2x-1$$

Whakawehea ngā taha e rua ki te -2x^{2}-x.

$$\frac{\left(-2x^{2}-x\right)a}{-2x^{2}-x}=\frac{-2x-1}{-2x^{2}-x}$$

Mā te whakawehe ki te -2x^{2}-x ka wetekia te whakareanga ki te -2x^{2}-x.

$$a=\frac{-2x-1}{-2x^{2}-x}$$

Whakawehe -1-2x ki te -2x^{2}-x.

$$a=\frac{1}{x}$$

Whakaoti mō a

$$\left\{\begin{matrix}a=\frac{1}{x}\text{, }&x\neq -\frac{1}{2}\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&x=-\frac{1}{2}\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xx=1-2axx+x\times 2-ax$$

Whakareatia te x ki te x, ka x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}=1-2axx+x\times 2-ax$$

Whakareatia te x ki te x, ka x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}=1-2ax^{2}+x\times 2-ax$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1-2ax^{2}+x\times 2-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}$$

Tangohia te 1 mai i ngā taha e rua.

$$-2ax^{2}+x\times 2-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1$$

Tangohia te x\times 2 mai i ngā taha e rua.

$$-2ax^{2}-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1-x\times 2$$

Whakareatia te -1 ki te 2, ka -2.

$$-2ax^{2}-ax=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1-2x$$

Pahekotia ngā kīanga tau katoa e whai ana i te a.

$$\left(-2x^{2}-x\right)a=\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}-1-2x$$

He hanga arowhānui tō te whārite.

$$\left(-2x^{2}-x\right)a=-2x-1$$

Whakawehea ngā taha e rua ki te -2x^{2}-x.

$$\frac{\left(-2x^{2}-x\right)a}{-2x^{2}-x}=\frac{-2x-1}{-2x^{2}-x}$$

Mā te whakawehe ki te -2x^{2}-x ka wetekia te whakareanga ki te -2x^{2}-x.

$$a=\frac{-2x-1}{-2x^{2}-x}$$

Whakawehe -1-2x ki te -2x^{2}-x.

$$a=\frac{1}{x}$$

Whakaoti mō f (complex solution)

$$f\in \mathrm{C}$$

$$x=-\frac{1}{2}\text{ or }\left(x=\frac{1}{a}\text{ and }a\neq 0\right)$$

Whakaoti mō f

$$f\in \mathrm{R}$$

$$x=-\frac{1}{2}\text{ or }\left(x=\frac{1}{a}\text{ and }a\neq 0\right)$$