4. Differentials

Differential

Definition

Given a function $y=f(x)$ the differential $dy=df$ of $f$ is given by

$dy=df=f^{\prime }(x)dx$

Note

If we set $x=x_0$ and $dx=△x$ we obtain the approximation

$△y=f(x_0+△x)-f(x_0)\approx dy$

Error

Definition

Consider a function $f(x)$ and an error $dx$ in the input value $x$ then:

Approximate

$df=f^{\prime }(x)dx$

Approximate

$\frac{df}{f(x)}=\frac{f^{\prime }(x)}{f(x)}dx$