# Application of Derivatives

Applications of derivatives are defined as the change (increase or decrease) in the quantity such as motion represents derivative. There are many important applications of derivative.

The concept of derivatives arose mainly as the result of many centuries of effort spent in drawing tangents to curves and finding the velocities of bodies in non-uniform motion. Derivatives are used widely in science, economics, medicine and computer science to calculate velocity and acceleration to explain the behavior of machinery and to estimate the decrease in water level as water is pumped out of a tank.

There are different applications of derivatives. It can be used in different situation.

## Theorems on Applications of Derivatives

Given below are some of the theorems on applications of derivatives:

### Rolle's Theorem

If a function f(x) is such that

1. f(x) is continuous on [a,b]
2. f(x) is differentiable on (a,b) and
3. f(a) = f (b)

then âˆƒ c âˆˆ (a,b) such that f'(c) = 0

### Geometrical Interpretation of Rolle's Theorem

Let AB be the graph of y = f(x) such that A = (a,f(a)) and B = (b,f(b))

1. f(x) is continuous between A and B.
2. f(x) has derivative between A and B i.e., there is a unique tangent at every point between A and B.
3. f(a) = f(b)

Then âˆƒ at least one point C between A and B such that the tangent at C is parallel to x-axis.

### Lagrange's Mean Value Theorem

If a function f (x) is such that

1. f(x) is continuous on [a,b]Â
2. f(x) is differentiable on (a,b)

then âˆƒ C âˆˆ (a,b) such that f'(C) = $\frac{f(b) - f(a)}{b - a}$

### Geometrical Interpretation of Lagrange's Mean Value Theorem

Let A(a, f(a)) and B(b, f(b)) be two points on the curve y = f (x) such that

1. f(x) is continuous between A and B
2. f(x) is differentiable between A and B

then there exists a point C(c, f(c)) between A and B such that the tangent at C is parallel to the chord AB.

Note that the point C is not unique.

### Approximation by Differentials

• Identify the function f(x).
• Write the values of x and Î”x
• Find $\frac{dy}{dx}$
• Find the actual changes in Î” dydx
• Calculate f(x+Î”x)

### Applications of Derivatives Summary

Generally, the following points are examined for tracing the curves

1. Passage through specific points
2. Points of intersection with the axes
3. Regions where the curve is increasing or decreasing
4. Region of existence
5. Symmetry about x - axis
6. Symmetry about y - axis