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.

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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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2018.07.15 Sunday (c) All rights reserved by ACHK