Editing Derivative (section)

=== Directional derivatives ===
{{Main|Directional derivative}}

If <math> f </math> is a real-valued function on {{tmath|1= \R^n }}, then the partial derivatives of <math> f </math> measure its variation in the direction of the coordinate axes. For example, if <math> f </math> is a function of <math> x </math> and {{tmath|1= y }}, then its partial derivatives measure the variation in <math> f </math> in the <math> x </math> and <math> y </math> direction. However, they do not directly measure the variation of <math> f </math> in any other direction, such as along the diagonal line {{tmath|1= y = x }}. These are measured using directional derivatives. Given a vector {{tmath|1= \mathbf{v} = (v_1,\ldots,v_n) }}, then the [[directional derivative]] of <math> f </math> in the direction of <math> \mathbf{v} </math> at the point <math> \mathbf{x} </math> is:{{sfn|Varberg|Purcell|Rigdon|2007|p=642}}
<math display="block"> D_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}.</math>
<!--In some cases, it may be easier to compute or estimate the directional derivative after changing the length of the vector.  Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector.  To see how this works, suppose that {{nowrap|1='''v''' = ''λ'''''u'''}} where '''u''' is a unit vector in the direction of '''v'''.  Substitute {{nowrap|1=''h'' = ''k''/''λ''}} into the difference quotient.  The difference quotient becomes:
:<math>\frac{f(\mathbf{x} + (k/\lambda)(\lambda\mathbf{u})) - f(\mathbf{x})}{k/\lambda}
= \lambda\cdot\frac{f(\mathbf{x} + k\mathbf{u}) - f(\mathbf{x})}{k}.</math>
This is ''λ'' times the difference quotient for the directional derivative of ''f'' with respect to '''u'''. Furthermore, taking the limit as ''h'' tends to zero is the same as taking the limit as ''k'' tends to zero because ''h'' and ''k'' are multiples of each other.  Therefore, {{nowrap|1=''D''<sub>'''v'''</sub>(''f'') = λ''D''<sub>'''u'''</sub>(''f'')}}.  Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.-->

If all the partial derivatives of <math> f </math> exist and are continuous at {{tmath|1= \mathbf{x} }}, then they determine the directional derivative of <math> f </math> in the direction <math> \mathbf{v} </math> by the formula:{{sfn|Guzman|2003|p=[https://books.google.com/books?id=aI_qBwAAQBAJ&pg=PA35 35]}}
<math display="block"> D_{\mathbf{v}}{f}(\mathbf{x}) = \sum_{j=1}^n v_j \frac{\partial f}{\partial x_j}. </math>