Calculus is a branch of Mathematics which deals with the methods of Derivation and Integration. These two
methods are the sub-branches of Calculus and are used to solve functions based on them. AP Calculus is
the Advanced Placement Examination in Calculus which is given in order to test a high school student in
Example 1: Find the derivative of the given function, f(x) = x5 + 2x3
Power rule of the Derivatives -> d f(xn)/dx = n*xn-1
Given function: x5 + 2x3
Applying the above formula for every term of the function, we get
Derivative of the given function -> d f(x)/dx= d (x5)/dx +2*d(x3)/dx
df(x)/dx = 5 *x5-1 + 2 * 3 * x3-1
d f(x)/dx = 5x4 + 6x2
Hence the derivative of the function f(x) =x5 + 2x3 is 5x4 + 6x2
Example 2: Calculate the definite integral value of the function, f(x) = 4x3 - 3x2 and ‘x’ ranging from
0 to 2.
∫xn dx= x(n+1)/ (n+1)
Apply the above formula for the given function, we get
∫(fx) dx = 4 *x3+1/(3 + 1) –3 *x2+1/(2 + 1)
∫f(x)dx = x4 –x3
First substitute x = 0 and x= 2 in the above answer.
When x= 0, ∫f(x) dx= 04 – 03 = 0
When x=2, ∫f(x)dx= 24– 23 = 8
Now subtract 8 - 0 = 8
Hence the definite integral of the given f(x) is 8.