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 :
Function | Derivative |
sinx | cosx |
cosx | -sinx |
tanx | sec2x |
cotx | -cosec2x |
secx | secx∙tanx |
cosec x | -cosecx∙cotx |
Derivatives Formula of Trigonometric Functions :
Function | Derivative |
sin-1x | 1/√(1-x2) |
cos-1x | -1/√(1-x2) |
tan-1x | 1/(1+x2) |
cot-1x | -1/(1+x2) |
sec-1x | 1(|x|∙(x2-1)) |
cosec-1x | –1(|x|∙(x2-1)) |
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).