# How to solve for K when setting the differential of a vega option with respect to K equal to 0?

The question is as follows:

Let $$v = S_0 \phi(d_1)\sqrt{T}$$. Solve the following equation for $$K$$. $$\frac{\partial v}{\partial K} = 0$$

By finding $$\frac{\partial v}{\partial d_1}$$ and $$\frac{\partial d_1}{\partial K}$$ I have used the following rule. $$\frac{\partial v}{\partial K} = \frac{\partial v}{\partial d_1}\cdot \frac{\partial d_1}{\partial K}$$ and achieved the following result:

$$\frac{\partial v}{\partial K} = \frac{S_0 d_1 \phi(d_1) \sqrt{T}}{\sigma K \sqrt{T-t} }$$

Now setting this equal to 0 I have been unable to solve for K. The furthest I've got is $$\frac{S_0 \sqrt{T}}{\sigma^2 K} \Big(\frac{\ln(S)-\ln(K)}{T-t} + r+\frac{\sigma^2}{2}\Big) = 0$$