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:

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

**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

**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:**

**Addition Rule of Derivatives:**

**Subtraction Rule of Derivatives:**$$\frac{d}{dx}(f-g)=\frac{df}{dx}-\frac{dg}{dx}$$

**Product Rule of Derivatives:**$$\frac{d}{dx}(fg)=f\frac{dg}{dx}+g\frac{df}{dx}$$

**Quotient Rule of Derivatives:**$$\frac{d}{dx}(\frac{f}{g})=\frac{g \frac{df}{dx}-f\frac{dg}{dx}}{g^2}$$

**Chain Rule of Derivatives:**

**(i)**$$\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)$$

**(ii)**$$\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}$$

**Examples:**

**Example 1: **