Derivative: definition, formulas, properties, and examples

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.


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:

$$\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}.$$

Sometimes the prime $’$ also denotes the derivative symbol.  Thus we have
$$f'(x)=\frac{d}{dx}(f(x))=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$$
Derivative at a point:  Let $f(x)$ be a function defined on an open interval $(c, d)$ containing $a$. The function $f(x)$ is said to be differentiable at $x=a$ if the limit
$$\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}$$
exists. This limit is called the derivative of $f(x)$ at the point $x=a,$ which is denoted by $f'(a)$ or $\frac{d}{dx}(f(x))|_{x=a}.$ So we have
$$f'(a)=\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}$$

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:

Let $f$ and $g$ be two differentiable functions of  $x.$ We will note down some properties of derivatives of $f$ and $g.$
Addition Rule of Derivatives: 
This is also known as the sum rule of derivatives.
Subtraction Rule of Derivatives: $$\frac{d}{dx}(f-g)=\frac{df}{dx}-\frac{dg}{dx}$$
This is also known as the difference rule of derivatives.
Product Rule of Derivatives: $$\frac{d}{dx}(fg)=f\frac{dg}{dx}+g\frac{df}{dx}$$
This is also known as the multiplication rule of derivatives.
Quotient Rule of Derivatives: $$\frac{d}{dx}(\frac{f}{g})=\frac{g \frac{df}{dx}-f\frac{dg}{dx}}{g^2}$$
This is also known as the division rule of derivatives.
Chain Rule of Derivatives: 
(i) $$\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)$$
(ii) $$\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}$$
The chain rule of derivatives is also sometimes called the derivative rule for the composition of functions.



Example 1: 

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