Derivative of Some Standard Functions

A derivative is a tool that allows us to understand how the relationship between two variables changes over time. Suppose you have two variables, one independent (x) and the other dependent (y). It is possible to calculate the derivative formula to find the relationship between the change in the dependent variable with respect to the change in the value of the independent variable expression. Using the derivative formula, you can find the slope of a line, the slope of a curve, and the change in one measurement with respect to another measurement in mathematical terms.

The derivative formula is one of the fundamental concepts in calculus, and the process of determining a derivative is referred to as differentiation in this context. With an exponent of “n,” the derivative formula is defined for a variable with an exponent of “x.” The exponent ‘n’ can be either an integer or a rational fraction, depending on the situation. As a result, the derivative can be calculated using the following formula:

  d/dx . xn = n. xn-1

The concept of derivatives can be extended to include functions of several real variables. After a suitable translation, the derivative is reinterpreted as a linear transformation, whose graph corresponds to the best linear approximation to the graph of the original function (after a suitable translation). Jacobian matrix: The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis that is determined by the selection of independent and dependent variables. In terms of partial derivatives with respect to the independent variables, it is possible to calculate it mathematically. The Jacobian matrix can be reduced to the gradient vector when dealing with a real-valued function with several variables.

Different types of derivatives

First and second-order derivatives are two types of derivatives that can be classified based on the order in which they are executed. These can be defined in the following manner.

First-order derivatives

The direction of the function is indicated by the first-order derivatives, which indicate whether the function is increasing or decreasing. The first derivative math, also known as the first-order derivative, can be thought of as a rate of change that occurs instantly. The slope of the tangent line can also be used to predict the result.

Second-order derivatives

Using second-order derivatives, we can get an idea of the shape of the graph corresponding to the given function and its derivatives. Concavity is a property of functions that can be used to classify them. The concavity of the given graph function can be divided into two categories, which are as follows:

• Concave up

• concave down

Derivation of the derivative formula

For simplicity, let us consider f(x) as a function whose domain contains an open interval around some point x0. Once the function is differentiable at point x0, it is said to be differentiable at point (x0), and the derivative of f(x) at point x0 is represented by the following formula:

f'(x)=lim ∆ₓ→0 ∆y/∆x

f'(x)= lim ∆ₓ→0 [f ((x0) + ∆x ) – f((x)0) /∆x ]

The derivative of the function y = f(x) is denoted by the symbol f′(x) or y′(x).

Furthermore, Leibniz’s notation is widely used to express the derivative of the function y = f(x) as df(x)/dx, which is equivalent to dy/dx.

List of derivatives formulas

Some of the most important derivative formulas are listed below, and they are used in various fields of mathematics, such as calculus, trigonometry, and other related fields. The differentiation of trigonometric functions is accomplished through the use of various derivative formulas, which are listed below. Almost all of the derivative formulas are derived from the differentiation of the first law.

Derivative formulas of elementary functions

• d/dx.xⁿ = n. x^n-1
• d/dx.k = 0 ( where k is a constant).
• d/dx.eˣ = eˣ
• d/dx.aˣ = aˣ.logₑ.a
• d/dx.log x = 1/x
• d/dx.√x = 1/(√2x )

Conclusion

If y = f(x), the derivative of a function of a variable x is the rate of change of y with respect to the rate of change of the variable x. 

A derivative is a tool that allows us to understand how the relationship between two variables changes over time. It is possible to calculate the derivative formula to find the relationship between the change in the dependent variable with respect to the change in the value of the independent variable expression. The derivative formula is one of the fundamental concepts in calculus, and the process of determining a derivative is referred to as differentiation in this context.

First and second-order derivatives are two types of derivatives that can be classified based on the order in which they are executed.

The direction of the function is indicated by the first-order derivatives, which indicate whether the function is increasing or decreasing.

Whereas, Using second-order derivatives, we can get an idea of the shape of the graph corresponding to the given function and its derivatives.