Den deriverte til en potensfunksjon

$$ $ \begin{array}{rcl}f\left(x\right) & =& x^2\\ & & \\ f^{\text{'}}\left(x\right) & =& 2x^{2-1}\\ & & \\ f^{\text{'}}\left(x\right) & =& 2x\end{array}$$$

$$\begin{gathered} $ \begin{array}{rcl}y\left(x\right) & =& x^5\\ & & \\ y^{\text{'}}\left(x\right) & =& 5x^{5-1}\\ & & \\ y^{\text{'}}\left(x\right) & =& 5x^4\end{array} \\ $ \end{gathered}$$

$$ $ \begin{array}{rcl}g\left(x\right) & =& 5x^7\\ & & \\ g^{\text{'}}\left(x\right) & =& 5·7x^{7-1}\\ & & \\ g^{\text{'}}\left(x\right) & =& 35x^6\end{array}$$$

$$ \begin{array}{rcl}y\left(x\right) & =& x^{-5}\\ & & \\ y^{\text{'}}\left(x\right) & =& -5x^{-5-1} \\ & & \\ y^{\text{'}}\left(x\right) & =& -5x^{-6} \mathrm{eller}\\ & & \\ y^{\text{'}}\left(x\right) & =& - \frac{5}{x^6}\end{array}$$

$$ \begin{array}{rcl}g\left(x\right) & =& 3x^{-4}\\ & & \\ g^{\text{'}}\left(x\right) & =& 3·\left(-4\right)x^{-4-1}\\ & & \\ g^{\text{'}}\left(x\right) & =& -12x^{-5} \mathrm{eller}\\ & & \\ g^{\text{'}}\left(x\right) & =& \frac{-12}{x^5}\end{array}$$

$$ \begin{array}{rcl}f\left(t\right) & =& -6t^3\\ & & \\ f^{\text{'}}\left(t\right) & =& -6·3t^{3-1}\\ & & \\ f^{\text{'}}\left(t\right) & =& -18t^2\end{array}$$

$$ $ \begin{array}{rcl}f\left(x\right) & =& x^0\\ & & \\ f\left(x\right) & =& 1\\ & & \\ f^{\text{'}}\left(x\right) & =& 0\end{array}$$$

$$\begin{gathered} $ \begin{array}{rcl}y\left(x\right) & =& \frac{1}{x^5}\\ & & \\ y\left(x\right) & =& x^{-5}\\ & & \\ y^{\text{'}}\left(x\right) & =& -5x^{-6}\\ & & \\ y^{\text{'}}\left(x\right) & =& -\frac{5}{x^6}\end{array} \\ $ \end{gathered}$$

$$ $ \begin{array}{rcl}g\left(x\right) & =& 5\sqrt{x}\\ & & \\ g\left(x\right) & =& 5x^{\frac{1}{2}}\\ & & \\ g^{\text{'}}\left(x\right) & =& 5·\frac{1}{2}·x^{\frac{1}{2}-1}\\ & & \\ g^{\text{'}}\left(x\right) & =& \frac{5}{2}{x}^{-\frac{1}{2}}\\ & & \\ g^{\text{'}}\left(x\right) & =& \frac{5}{2x^{{\displaystyle \frac{1}{2}}}}\\ & & \\ g^{\text{'}}\left(x\right) & =& \frac{5}{2\sqrt{x}}\end{array}$$$

$$ $ \begin{array}{rcl}z\left(t\right) & =& \frac{1}{\sqrt{t}}\\ & & \\ z\left(t\right) & =& t^{-\frac{1}{2}}\\ & & \\ z{\left(t\right)}^{\text{'}} & =& -\frac{1}{2}{t}^{-\frac{1}{2}-\frac{2}{2}}\\ & & \\ & =& -\frac{1}{2}{t}^{-\frac{3}{2}}\\ & =& -\frac{1}{2t^{{\displaystyle \frac{3}{2}}}}\\ & =& -\frac{1}{2t^{{\displaystyle \frac{2}{2}}}·t^{{\displaystyle \frac{1}{2}}}}\\ & & \\ & =& -\frac{1}{2t·\sqrt{t}}\end{array}$$$

Husk at her skal vi derivere med hensyn på $$ t$$.

$$ \begin{array}{rcl}f\left(t\right) & =& 3x^2\\ & & \\ f^{\text{'}}\left(t\right) & =& 0\end{array}$$

$$ \begin{array}{rcl}f\left(t\right) & =& 2t^3\\ & & \\ f^{\text{'}}\left(t\right) & =& 2·3t^2\end{array}$$