# Partial derivatives hold other things still

Subject: Deep Learning

Exam: MT2

Reader: https://deep-learning-study-room.vercel.app/#mt2/read/3/mt2-c3-s5

Canonical: https://deep-learning-study-room.vercel.app/read/dl/mt2/section/mt2-c3-s5/

For z = wx + b, the derivative with respect to w is x, because x is held fixed. The derivative with respect to b is 1. The derivative with respect to x is w. This is the entire local maths of a neuron’s weighted sum.

![Choose one changing parameter. Everything else is held fixed while you find its effect.](https://deep-learning-study-room.vercel.app/assets/figures/v_partial.svg)

Diagram: Choose one changing parameter. Everything else is held fixed while you find its effect.. Labels: Change one parameter. Hold the others fixed. | Change one parameter. Hold the others fixed. | For z = wx + b, choose which parameter you are changing. | Change w; keep x and b fixed | ∂z/∂w = x | If x = 3, the slope is 3. | Change b; keep x and w fixed | ∂z/∂b = 1 | Adding 1 to b adds 1 to z. | “Partial” identifies one chosen variable while the other variables stay fixed.

Choose one changing parameter. Everything else is held fixed while you find its effect. Tap the diagram to enlarge.



$$
\frac{\partial z}{\partial w}=x,\qquad \frac{\partial z}{\partial b}=1,\qquad \frac{\partial z}{\partial x}=w
$$



**Question:**  z = 3w + b, w = 2, b = 1. Find z, ∂z/∂w and ∂z/∂b.

### Given

z = 3w+b; w = 2; b = 1.

### To find

Find z, ∂z/∂w and ∂z/∂b.

### Formula — which one and why

**Partial derivatives.** Differentiate only with respect to the named variable; all other independent inputs and parameters remain fixed.

∂z/∂w reads “rate of change of z when w changes”. For z = wx+b, x is the coefficient of w, while b has coefficient 1.



$$
\frac{\partial z}{\partial w}=x
$$





$$
\frac{\partial z}{\partial b}=1
$$



**Yaad rakhne ka cue:** Neeche jis letter ka naam hai, sirf usko change samjho.

### Solution — step by step

**Step 1: calculate the function value.**



$$
z=3(2)+1=6+1=7
$$



**Step 2: differentiate with respect to w while b is fixed.**



$$
\frac{\partial z}{\partial w}=3+0=3
$$



**Step 3: differentiate with respect to b while w is fixed.**



$$
\frac{\partial z}{\partial b}=0+1=1
$$



**Final answer:** z = 7; weight derivative = 3; bias derivative = 1. A function value and its rate of change are different quantities.