# Properties of Differentials

• Functions: $$f$$, $$u$$, $$v$$
Argument (independent variable): $$x$$
Derivatives: $$y’\left( x \right),$$ $$f’\left( x \right)$$
Constant: $$C$$
Real numbers: $$A$$, $$\alpha$$
Small change in $$y:$$ $$\Delta y$$
Small change in $$x:$$ $$\Delta x$$
Differential of the function $$y:$$ $$dy$$
Differential of the independent variable $$x:$$ $$dx$$
1. Consider a function $$y = f\left( x \right)$$ and suppose that the independent variable gets an increment $$dx$$ at some point $$x$$. This increment is called the differential of the independent variable. The function $$y = f\left( x \right)$$ has a differential at the point $$x$$ if its increment can be represented as the sum of two terms:
$$\Delta y = f\left( {x + \Delta x} \right) – f\left( x \right) =$$ $$A\Delta x + \alpha ,$$
where the coefficient $$A$$ is independent of $$\Delta x$$ and the value of $$\alpha$$ has a higher order of smallness with respect to the increment $$\Delta x$$, i.e. $$\alpha /\Delta x \to 0$$ as $$\Delta x \to 0$$.

In the formula above, the principal linear part of the increment is called the differential of the function $$f\left( x \right)$$ at the point $$x$$ and denoted by $$dy = A\Delta x$$. In this expression the coefficient $$A$$ is equal to the value of the derivative $$f’\left( x \right)$$ at the point $$x.$$
2. The differential of the independent variable is equal to its increment: $$dx = \Delta x$$
3. The differential of a function is equal to the derivative of the function times the differential of the independent variable:
$$dy = df\left( x \right) =$$ $$f’\left( x \right)dx$$
4. Derivative as the quotient of two differentials
$$f’\left( x \right) = \large\frac{{dy}}{{dx}}\normalsize$$
5. The differential of a constant is zero: $$dC = 0$$
6. differential of the sum of two functions is equal to the sum of their differentials:
$$d\left( {u + v} \right) =$$ $$du + dv$$
7. differential of the difference of two functions is equal to the difference of their differentials:
$$d\left( {u – v} \right) =$$ $$du – dv$$
8. constant factor can be taken out of the differential:
$$d\left( {Cu} \right) = Cdu$$
9. Differential of the product of two functions
$$d\left( {uv} \right) = vdu + udv$$
10. Differential of the quotient of two functions
$$d\left( {{\large\frac{u}{v}}\normalsize} \right) = {\large\frac{{vdu – udv}}{{{v^2}}}\normalsize}$$