Second Derivative

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Derivatives play a very pivotal role in calculus which is an important branch in mathematics. Second derivative or the second order derivative is the derivative of the derivative of the function. Therefore the second derivative is the rate of change of the function which is changing itself. The second derivative of a function is also useful in graphing the given function. The second derivative of the function decides the concavity of the graph. If the value of the second derivative is positive then the graph is concave upwards and if the value of the second derivative is negative then the graph is concave downwards.

Example 1: Find the second derivative of the function f(x) = x3 + 2x2 - 6?

Solution: Given is the function f(x) = x3 + 2x2 - 6.

Step1: Find the first derivative of the function using the power rule.

The first derivative of the function f’(x) = 3 x2 + 4x.

The second derivative of the function is the derivative of f’(x).

This gives, f”(x) = 6 x + 4.

Hence the second derivative = 6 x + 4.

Example 2: Find the second derivative of the function f(x) = 2x3 - 4x2 + 10?

Solution: Given is the function f(x) = 2 x3 -4 x2 + 10.

Step1: Find the first derivative of the function using the power rule.

The first derivative of the function f’(x) = 6 x2 – 8 x.

The second derivative of the function is the derivative of f’(x).

This gives, f”(x) = 12 x - 8.

Hence the second derivative = 12 x - 8