Derivatives Formula
To find the derivative of different functions we need to learn different formulas. However, the basic rule to find the derivative by first principle is valid to all but as it can be too extensive sometimes so we refer to formulas for instant differentiation.
Some of the most important formulas for derivatives are discussed as follows:
Power Rule of Derivatives
Power Rule of Derivatives states that If a function y = f(x) = xn then its derivative
dy/dx = f'(x) = nxn-1
where n is an integer
For example, the derivative of f(x) = x3 is 3x(3-1) = 3x2.
Derivative of Exponential Function
The derivative of Exponential Function is listed below:
- d(ex)/dx = ex
- d(ax) = ax ln a
Where e is the Euler’s number and a is any real positive number.
Derivative of Logarithmic Function
The formula for derivatives of logarithmic functions is given below:
- d(ln x)/dx = 1/x
- d(logax)/dx = 1/(x ln a)
Where ln is the natural logarithm i.e., log with base e [Eular’s Number]
Derivatives of Trigonometric Functions
The derivatives of various trigonometric functions are listed below:
- d(sin x)/dx = cos x
- d(cos x)/dx = -sin x
- d(tan x)/dx = sec2x
- d(cot x)/dx = -cosec2x
- d(sec x)/dx = sec x tan x
- d(cosec x)/dx = -cosec x cot x
Derivative of Inverse Trigonometric Functions
If x = sin y then y = sin-1x is the inverse trigonometric function.
The derivative formula for inverse trigonometric function is given below:
- d(sin-1x)/dx = 1/√(1 – x2)
- d(cos-1x)/dx = -1/√(1 – x2)
- d(tan-1x)/dx = 1/(1 + x2)
- d(cosec-1x)/dx = -1/|x|√(x2 – 1)
- d(sec-1x)/dx = 1/|x|√(x2 – 1)
- d(cot-1x)/dx = -1/(1 + x2)
Derivatives | First and Second Order Derivatives, Formulas and Examples
Derivatives: In mathematics, a Derivative represents the rate at which a function changes as its input changes. It measures how a function’s output value moves as its input value nudges a little bit. This concept is a fundamental piece of calculus. It is used extensively across science, engineering, economics, and more to analyze changes.
Table of Content
- What are Derivatives?
- Derivatives Meaning
- Derivative by First Principle
- Types of Derivatives
- First Order Derivative
- Second Order Derivative
- nth Order Derivative
- Derivatives Formula
- Rules of Derivatives
- Derivative of Composite Function
- Chain Rule of Derivatives
- Derivative of Implicit Function
- Parametric Derivatives
- Higher Order Derivatives
- Partial Derivative
- Logarithmic Differentiation
- Applications of Derivatives
- Derivatives Examples
- Sample Problems on Derivatives
- Practice Problems on Derivatives
A derivative is a calculus tool that measures the sensitivity of a function’s output to its input. It is also known as the instantaneous rate of change of a function at a given position.
The derivative of a function with just one variable is the slope of the line that is tangent to the function’s graph for a given input value. In terms of geometry, the derivative of a function can be defined as the slope of its graph.
In this article we have covered Meaning of Derivatives along with types of derivatives, examples, formulas, applications and many more.
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