The concept of the derivative is the backbone of the theory of Calculus. Here we will learn the definition of the derivative of a function, its various formulas, and its properties with examples.
Table of Contents
Definition of the Derivative of a function:
Limit definition of derivative: Let f(x) be a function of the variable x. Then the limit definition of the derivative of f(x), denoted by $\frac{d}{dx}(f(x))$, is defined by the following limit:
$$\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}.$$
List of all derivative formulas:
The problems related to differential calculus can be easily solved if you have a complete list of derivative or differential formulas in your table. So we provide here a complete list of basic derivative formulas to help you.
Basic derivative formulas
1. Power rule of derivative: $\frac{d}{dx}(x^n)=nx^{n-1}$
2. derivative of a constant: $\frac{d}{dx}(c)=0$
3. derivative of an exponential: $\frac{d}{dx}(e^x)=e^x$
4. $\frac{d}{dx}(a^x)=a^x\log_e a$
5. derivative of a natural logarithm: $\frac{d}{dx}(\log_ex)=\frac{1}{x}$
6. derivative of a common logarithm: $\frac{d}{dx}(\log_a x)=\frac{1}{x\log_e a}$
Derivative formulas of trigonometric functions
1. derivative of sine function: $\frac{d}{dx}(\sin x)=\cos x$
2. derivative of cosine function: $\frac{d}{dx}(\cos x)=-\sin x$
3. derivative of tangent function: $\frac{d}{dx}(\tan x)=\sec^2x$
4. derivative of cotangent function: $\frac{d}{dx}(\cot x)=-\text{cosec}^2\, x$
5. derivative of secant function: $\frac{d}{dx}(\sec x)=\sec x \tan x$
6. derivative of cosecant function: $\frac{d}{dx}(\text{cosec}\,x)=-\text{cosec}\, x\cot x$
Derivative formulas of hyper-trigonometric functions
1. derivative of hyperbolic sine function: $\frac{d}{dx}(\sinh x)=\cosh x$
2. derivative of hyperbolic cosine function: $\frac{d}{dx}(\cosh x)=\sinh x$
3. derivative of hyperbolic tangent function: $\frac{d}{dx}(\tanh x)=\text{sech}^2x$
4. derivative of hyperbolic cotangent function: $\frac{d}{dx}(\coth x)=-\text{cosech}^2\, x$
5. derivative of hyperbolic secant function: $\frac{d}{dx}(\text{sech} x)=-\text{sech} x \tanh x$
6. derivative of hyperbolic cosecant function: $\frac{d}{dx}(\text{cosech}\,x)=-\text{cosech}\, x\coth x$
Derivative formulas of inverse trigonometric functions
1. derivative of inverse sine function: $\frac{d}{dx}(\sin^{-1} x)=\frac{1}{\sqrt{1-x^2}}$
2. derivative of inverse cosine function: $\frac{d}{dx}(\cos^{-1} x)=-\frac{1}{\sqrt{1-x^2}}$
3. derivative of inverse tangent function: $\frac{d}{dx}(\tan^{-1} x)=\frac{1}{1+x^2}$
4. derivative of inverse cotangent function: $\frac{d}{dx}(\cot^{-1} x)=-\frac{1}{1+x^2}$
5. derivative of inverse secant function: $\frac{d}{dx}(\sec^{-1} x)=\frac{1}{|x|\sqrt{x^2-1}}$
6. derivative of inverse cosecant function: $\frac{d}{dx}(\text{cosec}^{-1}\,x)=-\frac{1}{|x|\sqrt{x^2-1}}$
Some Properties of Derivative:
Examples:
Example 1:
This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.