Differentiability and Continuity: If a function is differentiable then it must be continuous. 

If a function is not continuous then it cannot be differentiable. 

A function may be continuous at a point and not differentiable there. 

Graphically: CONTINUOUS means connected.  

DIFFERENTIABLE means continuous, smooth and not vertical. 

Differentiation Patterns: 

\begin{aligned}D(k f(x)) &=k \cdot D(f(x)) \\D(f+g) &=D f+D g \\D(f-g) &=\text { Df }-\mathbf{D} g \\\mathbf{D}(f \cdot g) &=f \cdot D g+g \cdot \mathbf{D} f \\D(f / g) &=\frac{g \cdot D f-f \cdot D g}{g^{2}}\end{aligned}

The FINAL STEP used to evaluate f indicates the FIRST RULE to use to differentiate f.

To evaluate a derivative at a point, first differentiate and then evaluate.