# The Definition of a Derivative

Definition of Derivative: $\quad \mathbf{f}^{\prime}(\mathrm{x}) \equiv \lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \quad$ if the limit exists and is finite.

Notations For The Derivative: $\mathbf{f}^{\prime}(\mathbf{x}), \operatorname{Df}(\mathbf{x}), \frac{\mathbf{d} \mathbf{f}(\mathbf{x})}{\mathbf{d x}}$

Tangent Line Equation: The line $\mathbf{y}=\mathbf{f}(\mathbf{a})+\mathbf{f}^{\prime}(\mathbf{a}) \cdot(\mathbf{x}-\mathbf{a})$ is tangent to the graph of $\mathrm{f}$ at $(a,f(a))$.

Formulas: $D( constant )=0$

$D ( x^n ) = n*x^{n-1}$ (proven for $n$ = positive integer: true for all constants $n$)

$D ( sin(x) ) = cos ( x )$ and $D ( cos(x) ) = -sin( x )$

\begin{aligned} &\mathbf{D}(|\mathrm{x}|) = \left\{\begin{array}{lll} +1 & \text { if } \mathrm{x} > 0 \\ \text { undefined } & \quad \text { if } \mathrm{x} = 0 \\ -1 & \text { if } \mathrm{x} < 0 \end{array}\right. \end{aligned}

Interpretations of  $f '(x)$:

Slope of a line tangent to a graph
Instantaneous rate of change of a function at a point
Velocity or acceleration
Magnification factor
Marginal change