Overview

Linear Approximation

Derivatives can be used to get very good linear approximations to functions.  By definition, $$f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}.$$ In particular, whenever $x$ is close to $a$, $\frac{f(x)-f(a)}{x-a}$ is close to $f'(a)$, i.e., $\frac{f(x)-f(a)}{x-a}\approx f'(a)$.  Thus $f(x) - f(a)\approx f'(a) (x-a)$, so $$f(x) \approx f(a) + f'(a) (x-a)$$ whenever $x$ is close to $a$.

The function $L(x) = f(a) + f'(a) (x-a)$ is called the linearization of $f(x)$. The line $y=L(x)$ goes through $\big(a, f(a)\big)$ with slope $f'(a)$, so it is the line tangent to $y=f(x)$ at $\big(a, f(a)\big)$. You can see this by finding the line tangent to $f$ at $x=a$: 
$$
\begin{eqnarray*}
y-y_1&=&m(x-x_1)\\
y-f(a)&=&f'(a)(x-a)\\
y&=&f(a)+f'(a)(x-a).
\end{eqnarray*}
$$
Notice that $L(x)$ is thus the $y$-value on the tangent line, and $L(x)\approx f(x)$ near $x=a$. 
DO:  How are $L(a)$ and $f(a)$ related?  Think and draw a tangent line to see the answer, before looking below.

Differentials

Differentials are another name for the same idea.

Let $\Delta x = x-a$ be the change in $x$, and $\Delta f = f(x) - f(a)$ be the change in $f$. And let $dx=x-a$, and let $df= f'(a) dx$.

Our approximation is then $$\Delta f \approx df.$$
The actual changes $\Delta f$ and $\Delta x$ are the rise and run along the secant line between $P=\big(a, f(a)\big)$ and $R=\big(x, f(x)\big)$.
The differentials $df$ and $dx$ are the rise and run along the tangent line as we go from $a$ to $x$.