Formula of Derivatives

The slope of the function’s graph, or, more precisely, the slope of the tangent line at a point, is the derivative of the function. Its computation is based on the slope formula for a straight line, with the exception that curves require a limiting process. The slope is frequently described as the “rise” over the “run,” or the ratio of the change in y to the change in x in Cartesian terms. The slope formula for the straight line illustrated in the illustration is (y1-y0)/(x1-x0). If h is used for x1-x0 and f(x) for y, another method to represent this formula is 

Formulas of Derivatives : 

ddxxn= n.xn-1

ddxk= 0  , where k is constant

ddxex= ex 

ddxax= ax. loge. a where a >0

ddxlog x= 1x 

ddxlogae= 1xlogae 

ddxx= 12x 

Derivatives

The fluctuating rate of change of function with respect to an independent variable is termed as a derivative. When there is a variable quantity and the rate of change is not constant, the derivative is used. The derivative is a tool for determining the sensitivity of one variable (the dependent variable) to another one (independent variable).

Derivatives Formula of Trigonometric Functions :

Derivatives Formula of Trigonometric Functions :

Examples :

Example 1 :

Example 2:

Types of Derivatives

First Order Derivative

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

Second Order Derivative

To acquire an understanding of the shape of the graph for a given function, second-order derivatives are used. Concavity can be used to categorise the function.

Properties of Derivatives

Derivatives can be broken down into smaller portions to make evaluating given expressions easier. The terms are separated based on the operator used to split the expressions or functions, like plus (+), minus (-), or division (/).

Applications of Derivatives :

Derivatives are used in a variety of subjects, including science, engineering, physics, and others, in addition to arithmetic and real life. You should have studied how to find the derivative of many functions in earlier classes, such as trigonometric functions, implicit functions, logarithm functions, and so on. In this section, you’ll learn how to apply derivatives to mathematical concepts and real-world situations. This is also one of the most significant topics covered in Class 12 Math.

Derivatives are used in a variety of ways in mathematics, including:

• Minimum and Maximum Values

• Rate of Change of a Quantity

• Tangent and Normal to a Curve

• Linear Approximations

• Increasing and Decreasing Functions

Derivative in Finance

A derivative is a financial contract between 2 or more entities whose value is determined by an agreed-upon underlying financial asset or group of assets, like a security or an index.

Conclusion : 

We can find rates of change using derivatives. It enables us to determine the rate of change of velocity with respect to time, for example (which is acceleration). It also allows us to calculate the rate of change of x with respect to y, which is the gradient of the curve on a graph of y against x. There are a few simple criteria that can be used to quickly differentiate a variety of functions.

The derivative of y (with respect to x) is written dy/dx, pronounced “dee y by dee x” if y = some function of x (in other words, if y is equivalent to an expression including integers and x’s).