# Chain Rule of Differentiation

Consider the curve $y = f(x)$.

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$\displaystyle{\frac{d}{dx}}$ is an operator, meaning “the slope of the tangent of”. So the expression $\displaystyle{\frac{dy}{dx}}$, meaning $\displaystyle{\frac{d}{dx} (y)}$, is not a fraction.

In order words, it means the slope of the tangent of the curve $y = f(x)$ at a point, such as point $A$ in the graph.

The symbol $dx$ has no relation with the symbol $\displaystyle{\frac{dy}{dx}}$. It means $\Delta x$ as shown in the graph. In other words,

$dx = \Delta x$

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The symbol $dy$ also has no relation with the symbol $\displaystyle{\frac{dy}{dx}}$. It means the vertical distance between the current point $A(x_0, y_0)$, where $y_0 = f(x_0)$, and the point $C$ on the tangent line $y = mx + c$, where $m$ is the slope of the tangent line. In other words,

$dy = m~dx$

or

$\displaystyle{dy = \left[ \left( \frac{d}{dx} \right) y \right] dx}$

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The relationship of $\Delta y$ and $dy$ is that

$\displaystyle{\Delta y = \frac{dy}{dx} \Delta x + \text{higher order terms}}$

$\displaystyle{\Delta y = \frac{dy}{dx} dx + \text{higher order terms}}$

$\Delta y = dy + \text{higher order terms}$

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Similarly, for functions of 2 variables:

$\displaystyle{\Delta f(x,y) = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + \text{higher order terms}}$

$\displaystyle{df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy}$

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For functions of 3 variables:

$\displaystyle{df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz}$

$\displaystyle{\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} + \frac{\partial f}{\partial z}\frac{dz}{dt}}$

— Me@2018-07-15 09:30:29 PM

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