How to find Directional Derivatives?

In calculus, the directional derivative is a differentiable function which multivariate along a vector V with a point P. It represents nothing but the rate of change of a function which moves in the direction of V, through the point P. The directional derivatives are the generalization of the partial derivatives; where in one of the coordinate axis is always parallel in direction.

Mathematical Definition of Directional Derivatives
The directional derivative is defined as the rate of change of the scalar function f(x) = f(x1, x2, x3... xn) along the unit vector u = (u1, u2, u3….. un), in which the unit vector is defined by a limit in terms of h. Therefore the mathematical definition for the directional derivative can be written as,

 Du f(x) = lim h->0+ f(x+hu) – f(x) / h.

Derivation of Directional derivative

It is simple method to find the directional derivative formula which will be used to simplify the problems. Let us define a function that is made up of a single variable, such as g(z) which equals f(sum of x0 + az and y0 + bz). Here x0, y0, a, b are some numbers which are fixed. Now, z is the letter which does not represent a fixed number.  Then, by the actual definition, for a single variable function we know that, g’(z) equals the limit which tends to h->0 of g (sum of z and h) – g(z) whole divided by h.  Now applying the value of the derivative z as zero, we will get, g’(0) which equals limit which tends to h->0 of subtraction of g (h) and g(0) whole divided by h. Substituting the equation of g(z) mentioned earlier we will get g’(0) as lim h->0 of f(x0 + az, y0 + bz) – f(x0,y0) whole divided by h. Then this equation will be obviously equal to the Du f(x0, y0). Therefore we will finally come with a relationship of g’(0) which equals Du f(x0, y0). Now we can rewrite g(z) as g(z) = f(x, y) where x will be equal to x0 + az and y will be equal to y0 + bz. Then, applying the chain rule we will get the equation as,  g’(z) = dg/dz = Sum of (multiplication of df/dx and dx/dz ) and (multiplication of df/dy and dy/dz), which will be equal to sum of fx (x,y) a and  fy (x,y).After taking z=0, we will get x as x0 and y as y0 and applying these, we will get, g’(0) as sum of            fx (x0,y0) a and fy (x0,y0) b. Now equating this equation with the g’(z) equation of very first we derived, we will arrive with a formula as shown below:

Du f(x, y) = fx (x, y) a + fy (x, y) b.

This equation will provide a simple way of calculation than the previous limit based definition.
Directional derivatives are used in solving the matrix symmetrical valued functions; especially the second directional derivative and it plays a vital role in solving non linear type problems.