How to understand backpropagation using derivative
This is a continuation to this answer. The first part of that answer covered gradient descent. In short this algorithm finds a set of parameters which results in a minimum of a function. From the description of the algorithm you can see where the derivative of the function is needed.

How this applies to a neural network

Let's first consider a single neuron network. This is called the perceptron. We will use the sigmoid activation function. It has inputs $x$ and an associated output 

$\hat{y} = \frac{1}{1+e^{w^Tx+b}}$.

The model is parametrized by the weights $w$ and the biases. These are the model parameters we need to tune. So this single neuron takes a vector $x$ and using the weights vector and biases will output some value $y$. 

If the weights are randomly chosen then the output will also be random. This is of no use to us. We need to determine a way of calculating a measure of correctness or wrongness in order to know how we should change our weights. I will use a measure of how wrong we are and call this the cost 

$C = \frac{1}{2N} \sum_{i=0}^{N}(\hat{y} - y)^2$.

This is takes the squared difference between the predicted output $\hat{y}$ and the ground truth $y$. We will take the sum of this wrongness for all the instances that we try to predict, hence the summation. And we divide by $N$ to normalize the error. The additional division by 2 is simply to make the derivative easier to compute you will see soon.

So we want to minimize the cost function using gradient decent. Thus we need to take the derivative of the cost function with respect to the parameters we can tune, these being the weights $w$. We must thus compute $\frac{\partial C}{\partial w}$. Using calculus we can use chain rule in order to simplify our derivative as

$\frac{\partial C}{\partial w} = \frac{\partial C}{\partial \hat{y}}\frac{\partial \hat{y}}{\partial w}$

Our gradient descent calculation thus becomes

$w^{new} = w^{old} - \nu \frac{\partial C}{\partial \hat{y}}\frac{\partial \hat{y}}{\partial w}$

where $\nu$ is the learning rate.



Let's explicitly spell out this derivative

$\frac{\partial C}{\partial \hat{y}} = \hat{y} - y$

and, the second is the derivative of the activation function which we are using which is the sigmoid function.

$\frac{\partial \hat{y}}{\partial w} = \frac{1}{1+exp(w^Tx + b)} (1 - \frac{1}{1+exp(w^Tx + b)})$