Calculus is one of the most useful and important branch of Mathematics. Calculus is the study of functions and its different characteristics. Calculus has its own set of different formulas and methods used to solve various questions Calculus is applied to find the lengths, areas, volumes, graphs etc. of 2-D and 3-D shapes by using techniques of Differentiation and Integration. Calculus consists of two sub-branches which are called as Differential Calculus and Integral Calculus.
Example 1: Find the derivative of the function, f(x) = 5x3 + 2x2.
Solution: To find the derivative, the Power rule of the Derivatives says that:
d(xn)/dx = n * xn-1
First step: We can distribute the derivative to both the terms:
This gives; d (5x3 + 2x2)/ dx = [ d(5x3)/ dx ] + [ d(2x2)/ dx ]
Using the above formula, we get: (5 *3 * x3-1) + (2 * 2 * x2-1) = (15 * x3-1) + (4 * x2-1)
= (15 * x2) + (4 * x1) = (15 x2) + (4x)
Hence the derivative of f(x) is f’(x) = 15 x2 + 4x.
Example 2: Find the anti-derivative of the function f(x) = 20 - x?
Solution: Here the given function is f(x) = 20 - x.
Power rule states anti-derivative of xn is equal to xn+1/(n+1).
The anti-derivative of x is 1/2 x2.
Using the power rule the anti-derivative of 20 needs to be found.
20 can be written as 20x0.
Therefore, the anti-derivative of 20 x0 is 20x1.
Hence anti derivative F(x) = 20 x -1/2 x2.