# Simplifying $\left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z+\Delta z} - \left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z} = ?$

I am deriving a partial differential equation for wave in a string. $f$ represents displacement of the string at point $z$. I am stuck at a step.

Can anyone help me, how $\left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z+\Delta z} - \left.\dfrac{\partial f}{\partial z} \right\arrowvert_{z}$ can be reduced to $\dfrac{\partial^2 f}{\partial z^2} \Delta z$ ?

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Please avoid titles that are entirely in $\TeX$. Thanks. –  Ｊ. Ｍ. Oct 30 '11 at 15:33
@ J.M.: Ok. I got it. Will take care in future. –  orion Oct 30 '11 at 16:56

Let $g(z)=\frac{∂f}{∂z}|_z$. So we have $g(z+\Delta{z})-g(z)=\frac{∂g}{∂z}\Delta{z}=\frac{∂^2f}{∂z^2}\Delta{z}$.