Partial Derivatives of Sigmoid Function

Question

I am confused about the partial derivatives that I'm using in my deep learning model. The formula of the sigmoid is: $i_t = \sigma(W_{ii}*x_t+b_{ii} + W_{hi}*h_{t-1}+b_{hi})$

Notation meaning:

1. $\sigma$ = Sigmoid Function
2. $W_{ii}$ = Weight Input
3. $x_t$ = Input data
4. $b_{ii}$ = Bias Input
5. $W_{hi}$ = Weight Hidden
6. $h_{t-1}$ = Hidden State previous Time Step
7. $b_{ii}$ = Bias Hidden

I've been searched that the regular sigmoid function partial derivatives is: $\frac{d}{d_x} = \sigma(x) . (1-\sigma(x))$

Then, I'm trying to do a partial derivative of the $i_t$ formula w.r.t $W_{ii}$, my current answer is: $\frac{d}{dw_{ii}} = \sigma(x_t) . (1-\sigma(x_t))$

Am I do wrong? If I do wrong, can someone help me to correct my answer?

UPDATE: I've been try again and this is my step right now:

1. Assume that $z = W_{ii}*x_t+b_{ii} + W_{hi}*h_{t-1}+b_{hi}$
2. Then the $i_t$ formula will be: $i_t = \sigma(z)$
3. Partial derivative will be: $\frac{d}{d_z}i_t = \sigma(z) . (1-\sigma(z))$
4. On the other hand, the partial derivative of the $z$ is: $\frac{d}{dw_{ii}}z = x_t $
5. Finally, the final partial derivative formula is: $\frac{d}{dw_{ii}}i_t = \sigma(z) . (1-\sigma(z)) . x_t$

But, I don't know whether it's correct or wrong. Can someone correct my updated answer? Thank you in advance