Introduction
Differential calculus can be considered as mathematics of motion, growth and change where there is a motion, growth, change. Whenever there is variable forces producing acceleration, differential calculus is the right mathematics to apply.
Increasing and Decreasing Functions
Let f be a function defined on an interval I and let x1 and x2 be any two points on I.
(i) f is said to be increasing in the interval I,
Maxima and Minima
A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) 
f(x) for all x
I
The number f(a) is called the local maximum of f(x). The point a is called the point of maximum.
Rolle's Theorem and Mean Value Theorem
Rolle's Theorem: Let f be a real valued function in [a,b] such that
- f is continuous in [a,b].
- f is differentiable in (a,b).
Langrange's Mean Value Theorem
Let f be real valued function in [a,b] such that,
1. f is continuous in [a,b].
2. f is differentiable in (a,b).
ApproxiMations by Differentials
Let y = f (x) be a differentiable function of x, errors in x and y are denoted by dx and dy, we have
\ Error in y = f ' (x) dx.
Rate of Change of Quantity
If a quantity y varies with respect to another quantity 'x' satisfying some rule y = f(x), in other words if y is a function x, then 
represents the rate of change of y with respect to x
Tangents and Normals
Let (dy/dx)(x1,y1) = m be the slope of the tangent at (x1, y1) on the curve y=f(x).
1. Equation of the tangent at (x1,y1) is 
2. Equation of the normal at (x1,y1) is 
Curve Sketching
The graph of a given function gives a visual presentation of the behaviour of the function involving a study of symmetry, rise and fall, region of existence, passage through specific points etc.
The following points are helpful to trace the curve.
a) Symmetry about x - axis: If the equation of the curve remains unaltered when y is replaced by -y, then the curve is symmetrical about x-axis.
b) Symmetry about y-axis: If the equation of the curve remains unaltered when x is replaced by -x then the curve is symmetrical about y-axis.
c) Symmetry about y = x: If the equation of the curve remains unaltered if x and y are interchanged, then the curve is symmetrical about y = x.
Summary
1. If m = 0 the tangent at (x1, y1) is parallel to x-axis.
2. If m1 = m2 the curves touch each other.
3. Rolle's theorem
4. Langrange's Mean Value theorem
5. Maxima and Minima
Conclusion
In this chapter we have learnt the application of derivatives to rate measure, also we have used the geometrical measurement of dy/dx to find the equations of the tangent and normal to a curve at any point on the curve, angle of intersection of the curves.
