Differentiation is a concept of calculus that refers to the process of determining the rate of change of a function or the computation of its derivative. Algebraically speaking, differentiation is the process of determining a set of functions’ derivatives.
A function’s derivative is the slope or gradient of the curve (graph) at any given moment in time. The gradient of a curve at a particular location is equal to the gradient of the tangent drawn to that curve at the point where the gradient is determined.
The gradient of a nonlinear curve can change at different places along the axis when the curve is not linear. This makes it very hard to calculate the gradient or slope at any given location. It is necessary to utilize differential equations to compute the gradient of a curve and to determine the instantaneous rates of change from one point to another.
Differential techniques are used to solve various vital concepts, for instance, there is a method of differentiation known as the “chain rule” that is used to distinguish between composite functions.
“The change in a property with respect to a unit change in another property” is another definition used to describe the term derivative. The calculation of instantaneous velocity is one of the applications of differentiation in practical life.
The functions are categorized generally into two types, i.e explicit functions and implicit functions. And to solve implicit functions, implicit differentiation is utilized. Also, it is possible to define an algebraic relationship between variables by using implicit functions.
Explicit functions are in the form y =… while explicit functions could be represented as implicit ones. Let’s say that you have y = 2x – 5. 2x – y = 5 is an implicit form of this expression that may be used to write this function.
While you might easily convert this specific equation into an explicit form, it can be difficult, if not impossible, to solve for y, which means you can’t use any of the “normal” methods of differentiation to solve for this particular equation. In such a situation the implicit differentiation calculator is used to determine the derivatives of the implicit functions.
The derivatives of a function of several variables in which all variables save the variable of interest are maintained constant during the differentiation process, according to the definition are the partial derivatives.
In order to denote the partial derivative of f with respect to variable x in mathematical terms, consider the function f(x,y) to be a two-variable function. In the case of keeping the variable y constant while differentiating the function f with regard to the variable x, using the principles and formulae of differentiation, we would write
Then, if we change the variable x as constant while differentiating the function (assuming that f is differentiable) with respect to the variable y. The partial derivative of “f” with respect to y is represented by the formula
The directional derivative is the rate at which a function changes at any specific location in a specified direction when it is applied to the function. Any derivative can be represented as a vector form as the directional derivative.
While characterizing the instantaneous pace at which the function changes it also broadens the scope of the partial derivative concept. Mathematically, the directional derivative of a scalar function f(x) = f(x1,x2,…,xn) is defined as a function of the following form:
▽uf = limh→0[f(x+hv)-f(x)]/h
This satisfies the statement, the directional derivative is a rate of change of a multivariable function in any direction, and it may be taken in any direction. In this expression, v is a vector along which the directional derivative of f(x) may be calculated.
In certain cases, v is confined to a unit vector however, the vector v is denoted by
v = (v1, v2, v3,…, vn)