# Q10 – Backpropagation

Contributed by Matthew Zak.

1. Create a very simple graph(circuit) given with $f(x_1,x_2,x_3,x_4)=x_1x_2+x_3x_4$ and compute all the derivatives of f with respect to inputs $(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\frac{\partial f}{\partial x_3},\frac{\partial f}{\partial x_4})$ using a chain rule ($\frac{\partial f}{\partial x}=\frac{\partial f}{\partial q}\frac{\partial q}{\partial x}$).
2. Show how will the gradient of $f$ with respect to $x_1$ change when we increase the input $x_2$ by $\Delta h$.
3. Having a function $g(f(x1, x2, x3, x4))$ where $f$ is given by the function above and $g(t) = \sigma(t)$ is is a sigmoid function, compute the derivative of $g$ with respect to input $x_1(\frac{\partial g}{\partial x_1})$.

## 4 thoughts on “Q10 – Backpropagation”

1. Théo Rubenach says:

I am not sure about the meaning of the word “circuit”, as I do not see how to build a graph with a closed loop from f. Does someone have an explanation ?

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2. gagnonlg says:

For 2., do you mean the _derivative_ of f with respect to x1, or the gradient of f with respect to _x_?

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• I think he talks about the component x1 so I considered the derivative of f w.r.t x1

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• Since the gradient of $f$ with regards of $x_1$ is a function of $x_2$. I think he means what is the impact on the gradient.

I think the answer is $\Delta h$.

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