Approximations using derivatives

Question

I came across the following definitions in my textbook:

1. The differential of $x$, denoted by $dx$, is defined by $dx = \Delta x$
2. The differential of $y$, denoted by $dy$, is defined by $dy=f'(x) dx$ or $dy = (\frac{dy}{dx})\Delta x$

I understood the first part. However, the second part doesn't make intuitive sense to me. What is the intuitive explanation for the second definition?

Answer 1

Think of derivative as slope of the tangent line to the graph of $y=f(x)$ at the point $(x,f(x))$

If you approximate your function with its tangent line, then $$m=f'(x)=\frac {dy}{dx}$$ where $dy$ is the linear approximation to $\Delta y$ which is the actual change in $y$

As you see,$$ f'(x)=\frac {dy}{dx}$$ could be written as $$dy=f'(x) dx$$

Answer 2

If $\Delta x$ is sufficiently close to $0$, then $\frac{\Delta y}{\Delta x}\approx\frac{dy}{dx}$ where $\frac{dy}{dx}$ denotes the derivative of $f(x)$ at a particular point. That's quite an obvious fact. If we multiply both sides by $\Delta x$ (we can do that because $\Delta x\ne 0$), we'll get the following:

$$\Delta y\approx\frac{dy}{dx}\Delta x.$$

The next step is purely a notational thing. Replace $\Delta y$ with $dy$, $\Delta x$ with $dx$, $\approx$ with $=$ and you will get:

$$dy=\frac{dy}{dx}dx.$$

This is called the differential of $y$. That's how it's defined.