The definition of a differential is based on Figure 3.4.1, the fundamental diagram of differential calculus. Points A and C are on the line tangent to the red curve at point A. In the right-triangle ΔABC, the angle the hypotenuse (the tangent line) makes with the horizontal is $\mathrm{\θ}$, and by the definition of the derivative at A, $f\prime \=\tan \left(\mathrm{\θ}\right)$. Consequently,

Thus $\mathrm{df}$, the differential of $f\left(x\right)$, is defined as the derivative $f\prime \left(x\right)$ times $\mathrm{dx}$, an increment (large or small) in $x$, the independent variable.

Figure 3.4.1 then suggests that $\mathrm{df}$ is an approximation to $\mathrm{\Δ}$$f$, the exact change in $f$ as the independent variable changes from $x$ to $x\+\mathrm{dx}$.

This idea is captured in the notation $\mathrm{\Delta } f\=f\left(x\+\mathrm{dx}\right)-f\left(x\right)\doteq \mathrm{df}$ 

Isolating $f\left(x\+\mathrm{dx}\right)$ leads to the linear approximation $f\left(x\+\mathrm{dx}\right)\doteq f\left(x\right)\+\mathrm{df}$

$f\left(x\right)$ is continuous in $\left[a\,b\right]$ 

$f\left(x\right)$ is differentiable in $\left(a\,b\right)$ 

At least one $c$ exists in $\left(a\,b\right)$ for which $f\prime \left(c\right)\=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ 

Approximate $\sqrt{17}$ by using the differential of the function $f\left(x\right)\=\sqrt{x}$.  

How accurate is this approximation?

Apply the Mean Value theorem to the function $f\left(x\right)\=x e^{-x}\,0\le x\le 3.$ 

Determine $c$ from first principles.