## Derivative Rules

The rate of change of one quantity with respect to some another quantity has a great importance.

The rate of change of a quantity ‘y’ with respect to another quantity ‘x’ is called the **derivative** or differential coefficient of y with respect to x.

The **Derivative** means the slope of a function at any point.

### Some Standard Differentiation Formulae

**(1) Differentiation of some common functions:**

**(2) Differentiation of algebraic functions:**

In particular

**(3) Differentiation of trigonometric functions:**

**(4) Differentiation of logarithmic and exponential functions:**

**(5) Differentiation of inverse trigonometrical functions:**

**(6) Differentiation of hyperbolic functions:**

**(7) Suitable substitutions**

### Rules for Differentiation

Let f(x), g(x) and u(x) be differentiable functions

- If at all points of a certain interval, f'(x) = o, then the function f(x) has a constant value within this interval.
**Chain rule**

**(i) Case I:**if y is a function of u and u is a function of x, then derivative of y with respect to x is

**(ii) Case II:**If y and x both are expressed in terms of t, y and x both are differentiable with respect to t, then

**Sum and difference rule: Using linear property**

**Product rule**

**Scalar multiple rule:**

**Quotient rule:**

Provided g≠0.