> [!summary]
This note shows the shortcuts for the derivative rules
>
**Key Table:**
>
>
| Rule | Derivative Shortcut |
| ---------------------- | ------------------------------------------------------------------------------------------------- |
| Power Rule | $\frac{d}{dx }x^n = nx^{n-1}$ |
| Sum/ Difference Rule | $\frac{d}{dx}[f (x)\pm g (x)] = \frac{d}{dx}f (x)\pm\frac{d}{dx}g (x)$ |
| Constant Multiple Rule | $\frac{d}{dx}[f (x)-g (x)] = \frac{d}{dx}f (x)-\frac{d}{dx}g (x)$ |
| Product Rule | $\frac{d}{dx}[f (x)\cdot g (x)] = f' (x) g (x)+f (x) g' (x)$ |
| Chain Rule | $\frac{d}{dx}[C\cdot F (x)] = C\cdot F' (x)$ |
| Quotient Rule | $\frac{d}{dx} \left[ \frac{f (x)}{g (x)} \right] = \frac{f' (x) g (x)-f (x) g' (x)}{[g (x)]^2}$ |
>[!info]+ Read Time
**⏱ 3 mins**
> [!warning] Assumptions
These are rules and proof of [[Derivative|derivatives]], so assume the following:
> - The notation for a derivative of x is $\frac{d}{dx}(x)$
# Power Rule
$
\frac{d}{dx }x^n = nx^{n-1}
$
> [!info]- Proof
![[Proof of Derivative Power Rule]].
# Sum Rule
$
\frac{d}{dx}[f(x)+g(x)] = \frac{d}{dx}f(x)+\frac{d}{dx}g(x)
$
> [!info]- Proof
![[Proof of Derivative Sum Rule]]
# Difference Rule
$
\frac{d}{dx}[f(x)-g(x)] = \frac{d}{dx}f(x)-\frac{d}{dx}g(x)
$
> [!info]- Proof
![[Proof of Derivative Difference Rule]]
# Constant Multiple Rule
Note that $C$ is a [[Real Numbers|real number]].
$
\frac{d}{dx}[C\cdot F(x)] = C\cdot F'(x)
$
> [!info]- Proof
![[Proof of Derivative Constant Multiple Rule]]
# Product Rule
$
\frac{d}{dx}[f(x)\cdot g(x)] = f'(x)g(x)+f(x)g'(x)
$
> [!info]- Proof
![[Proof of Derivative Product Rule]]
# Chain Rule
$
\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g(x)
$
> [!info]- Proof
![[Proof of Derivative Chain Rule]]
# Quotient Rule
$
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}
$
> [!info]- Proof
![[Proof of Derivative Quotient Rule]]
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