Special Methods of Differentiation

In Calculus, a derivative is the instantaneous rate of change of a function with respect to one of its variables, and in geometry, a derivative is the geometric meaning of a derivative. The derivative of a function is well defined according to the first derivative principle. This is defined as the slope of the graph at the point where the first derivative of a function is defined as the first derivative of a function. The second derivative of a function at a point, on the other hand, is a measure of the degree to which the graph is deflected from the tangent at the point of contact.

Differentiation is defined as the rate at which one quantity changes in relation to another quantity. The rate of change of distance with respect to time is used to calculate the speed of a vehicle. The speed measured at each instant is not the same as the average speed calculated in the previous step. The term “speed” is synonymous with the term “slope,” which is nothing more than the instantaneous rate of change of distance over a given period of time.

Differentiation is defined as the relationship between a small change in one quantity and a small change in another quantity that is dependent on the first quantity. Calculus’ differentiation of a function is a crucial concept, and it is one of the most important concepts to understand. If y = f(x) is a differentiable function, then the differentiation is represented as f'(x) or dy/dx in the expression f(x).

Logarithmic Differentiation Functions:

Whenever a function is both the product and quotient of functions, as in y = f₁(x).f₂(x)…../g₁(x).g₂(x)……  Alternatively, if a function is expressed as an exponent of one function over another, as in [f(x)] g(x), we take the logarithm of the function f(x) (to base e) and then differentiate it.

For example, if y = x^x, then log y = xlogx

1/y.dy/dx = logx + 1

dy/dx = y. (logx + 1)

= x^x (logx + 1).

Parametric Differentiation:

There are times when, rather than defining a function explicitly or implicitly, we define it by referring to a third variable as the function definition. When a function y(x) is represented by a third variable, which is known as the parameter, this representation is referred to as a parametric representation. A relationship between x and y can be expressed in the form x = f(t), and a relationship between x and y can be expressed in the form y = g(t), where t is the parameter.

Rules for resolving problems involving the derivatives of functions expressed in parametric form include those that follow:

Step I) First, we write the given functions x and y in terms of the parameter t, which is the first step.

Step ii) Determine the answer using differentiation.

          dy/dt and dx/dt

Step iii) we will use the formula for solving functions in parametric form, which is as follows:

dy/dx = dy/dt| dx/dt

Step iv) Substitute the values of the variables

dy/dt and dx/dt

and simplify to obtain the result.

Parametric Equation:

They are known as parametric equations when a group of quantities of one or more independent variables is formed as functions of one or more of the independent variables. These are used to represent the coordinates of a point for any geometrical object such as a curve, a surface, or any other geometrical object, and the equations of these objects are referred to as a parametric representation of that particular object in mathematics.

For the most part, parametric equations have the following form: 

x=cost 

y=sint

As shown in the equation (x, y) = (cost, sint), the unit circle is represented by a parametric form in which t is the parameter and (x, y) are two points on the unit circle.

Applications:

Parametric functions are primarily used in the integration of various types of functions when the given function is represented in the complex form, as in the following example. A parameter t is used to substitute for a portion of the given function in these situations. Some of the other applications include graphs of various functions and equations involving differentiation, among other things.

Conclusion:

Techniques of Differentiation investigates a number of different rules, including the product, quotient, chain, power, exponential, and logarithmic rules, as well as other variations.

The derivative of a function is well defined according to the first derivative principle. This is defined as the slope of the graph at the point where the first derivative of a function is defined as the first derivative of a function.

Differentiation is defined as the rate at which one quantity changes in relation to another quantity.

Differentiation is defined as the relationship between a small change in one quantity and a small change in another quantity that is dependent on the first quantity. If y = f(x) is a differentiable function, then the differentiation is represented as f'(x) or dy/dx in the expression f(x). If a function is expressed as an exponent of one function over another, as in [f(x)] g(x), we take the logarithm of the function f(x). 

There are times when, rather than defining a function explicitly or implicitly, we define it by referring to a third variable as the function definition. When a function y(x) is represented by a third variable, which is known as the parameter, this representation is referred to as a parametric representation.