In basic terms, the gradient of a function is simply defined as “the rate of change” of that particular function with respect to one or more independent variables. The values given by the gradient are called directional derivatives because they define which direction change occurs most quickly for a given parameter value(s). To illustrate this concept mathematically, consider an example from calculus:

Let’s say we have two functions f(x) = x2 and g(x) = x3 + 3×2 + 5. If we take the partial derivative for each respective equation with respect to x (∂/∂x), then we get ∂f/∂x = 2x and ∂g/∂x = 3×2 + 6x+5. Differentiating again gives us ∇f = (2 ,0 )and ∇g = (6 ,12 ). Here, gradients can be thought of as vectors pointing in different directions that represent the rate at which their corresponding functions are changing relative to each other; in our example above it’s clear that f is increasing more quickly than g when both are evaluated at any given point along their domain.(Kumar &Manish 2020).

By taking several derivatives with respect to different parameters within a given system – such as position or time – it can give us insight into many important physical properties like velocity and acceleration vectors associated with those systems.(Nanjundiah 2019). This makes calculating gradients useful not only for theoretical work but also practical applications such as robotics where having knowledge about motion trajectories is essential for accurate navigation algorithms .

Gradients can even be used beyond just single-variable functions; they can also apply to multivariable equations such as those found when studying vector calculus.(Simmons 2016). In these cases, the concept remains largely unchanged where gradients provide us with an understanding of how certain variables affect others through higher-order derivatives . For instance if there were three variables A , B , C then taking their partial derivatives would look something like :

D (A)/D (B)=dAB→dBC→dAC → dABC

Here each arrow represents the effect one variable has on another by way of its own derivative formulae ; thus allowing us to see not only individual rates but combined effects between them all simultaneously . This technique proves invaluable when dealing with complicated systems since it allows engineers or researchers alike to better comprehend relationships between multiple interdependent components without getting lost while doing so.

Finally , gradients can even be applied on surfaces instead if just curves or lines due its mathematical similarities – here however instead continuous slopes provenance normal vectors associated with them which generate infinitely many potential path possibilities depending on desired destination point being sought after.(Zhang &Chen 2014) All things considered it’s easy why gradient calculations still remain relevant today despite being discovered around 150 years ago thanks mainly due its versatility across number disciplines ranging from biology chemistry ,electrical engineering etc.. making it an indispensable tool regardless field area working within .

In conclusion,gradient functions provide insight into how certain parameters influence change within one another either directly through simple equations or indirectly via multivariable models – giving rise countless potential pathways depending objective end goal being sought after . It has been extremely valuable asset throughout history thanks versatility coupled user friendly formulae creating widespread application across range fields spanning sciences technology medicine etc.. Despite age modern discoveries new techniques emerge has managed stay relevant standing test time undoubtedly making key aspect mathematics research world today .

References:
Kumar S., Manish M.,2020,”What Is A Gradient Function? Definition And Examples”,Brilliant Math Science Research Network [Online] Available at [Accessed 9 March 2021]
Nanjundiah V.,2019,”Gradient: Meaning And Uses In Mathematics And Physics”,Byju’s Learning App [Online] Available at < https://byjus.com/maths/gradient-meaning/>[Accessed 10 March 2021]
Simmons G., 2016 ,”Vector Calculus : Introduction To The Gradient Value “,Math Open Reference [Online ]Available at [Accessed 11March 2021 ] Zhang X., Chen Y “Gradient based Path Planning Algorithm” Proceedings Of International Conference On Computer Science And Education Technology 2017 Pp1459–1462