# Application of Derivatives: Rate of Change of a Quantity, Increasing and Decreasing Functions (For CBSE, ICSE, IAS, NET, NRA 2022)

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## What Are Derivatives?

- Derivatives are referred to the situation when a quantity varies with respect to another quantity , fulfilling a condition and represents the change of rate of with respect to and where represents the rate of change of with respect to
- Derivatives have various important applications in Mathematics such as:
- Rate of Change of a Quantity
- Increasing and Decreasing Functions
- Tangent and Normal to a Curve
- Minimum and Maximum Values
- Newton՚s Method
- Linear Approximations

### Rate of Change of a Quantity

This is the general and most important application of derivative. For example, to check the rate of change of the volume of cube with respect to its decreasing sides, we can use the derivative form as . Where represents the rate of change of volume of cube and represents the change of sides of the cube.

### Increasing and Decreasing Functions

To find that a given function is increasing or decreasing or constant, say in a graph, we use derivatives. If f is a function which is continuous in and differentiable in the open interval ,

- is increasing at if for each
- is decreasing at if for each
- is constant function in [p, q] , if for each

### Minimum and Maximum Values

It is important to understand the difference between the two types of minimum/maximum (collectively called extrema) values for many of the applications in this chapter and so we use a variety of examples to help with this.

### The Mean Value Theorem

In this section we will give Rolle՚s Theorem and the Mean Value Theorem. With the Mean Value Theorem, we will prove a couple of very nice facts.

### Tangent and Normal to a Curve

- Tangent is the line that touches the curve at a point and doesn՚t cross it, whereas normal is the perpendicular to that tangent.
- Let the tangent meet the curve at

Now the straight-line equation which passes through a point having slope m could be written as;

We can see from the above equation, the slope of the tangent to the curve and at the point , it is given as at . Therefore,

**Equation of the tangent** to the curve at can be written as:

**Equation of normal** to the curve is given by;