Derivatans definition

Detta blir ganska horribelt men använd att

$$\begin{gathered} (x+h{)}^{-3/2}-x^{-3/2}=\frac{1}{(x+h{)}^{3/2}}-\frac{1}{x^{3/2}}=\frac{x^{3/2}-(x+h{)}^{3/2}}{x^{3/2}(x+h{)}^{3/2}}= \\ \frac{x^3-(x+h{)}^3}{x^{3/2}(x+h{)}^{3/2}\left(x^{3/2}+(x+h{)}^{3/2}\right)}=\frac{x^3-(x^3+3x^{2}h+3x{h}^2+h^3)}{x^{3/2}(x+h{)}^{3/2}\left(x^{3/2}+(x+h{)}^{3/2}\right)}= \\ -\frac{3x^{2}h+3x{h}^2+h^3}{x^{3/2}(x+h{)}^{3/2}\left(x^{3/2}+(x+h{)}^{3/2}\right)} \end{gathered}$$

Vilket sedan ger att

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{(x+h{)}^{-3/2}-x^{3/2}}{h}=\underset{h\to 0}{\lim }-\frac{{\displaystyle 3x^{2}h+3x{h}^2+h^3}}{{\displaystyle h{x}^{3/2}(x+h{)}^{3/2}\left(x^{3/2}+(x+h{)}^{3/2}\right)}}= \\ \underset{h\to 0}{\lim }-\frac{{\displaystyle 3x^2+3xh+h^2}}{{\displaystyle x^{3/2}(x+h{)}^{3/2}\left(x^{3/2}+(x+h{)}^{3/2}\right)}} \end{gathered}$$

Sedan beräknar du detta gränsvärde.

Tack för hjälpen:)