Derivatives of Polynomial Functions

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Derivatives are used in Calculus to measure the rate of change of a function with respect to a variable. The use of derivatives is very important in Mathematics. It is used to solve many problems in mathematics like to find out maxima or minima of a function, slope of a function, to tell whether a function is increasing or decreasing. If a function is written as y = f(x) and we want to find the derivative of this function then it will be written as dy/dx and can be pronounced as the rate of change of y with respect to x.

The derivative of a polynomial function

To calculate the derivative of a polynomial function, first, you should know the product rule of derivatives and the basic rule of the derivative.

Product rule of derivative

$\frac{\partial (x^{n})}{\partial x} = n\times x^{n-1}$

(Here n can be either positive or negative value)

Understand in this way: The old power of the variable is multiplied with the coefficient of the variable and the new power of the variable is decreased by 1 from the old power.

Example: Find the derivative of x³?

Solution:

Let y = x³

$=> \frac{\partial y}{\partial x} = 3\times x^{3-1} = 3x^2$

Some basic rules of derivative

If y = c f(x)

$\frac{\partial y}{\partial x} = c\frac{\partial (f(x))}{\partial x}$

If y = c

$\frac{\partial y}{\partial x} = 0$

$If \ y= f_{1}(x)\pm f_{1}(x)$

$\frac{\partial y}{\partial x} = \frac{\partial (f_{1}(x))}{\partial x}\pm \frac{\partial (f_{1}(x))}{\partial x}\\$

Example 1: Find the derivative of 4x³+ 7x?

Solution:

Let y = 4x³+ 7x

$\frac{\partial y}{\partial x} = \frac{\partial (4x^{3})}{\partial x}+\frac{\partial (7x)}{\partial x} \\ \frac{\partial y}{\partial x} = 4\times 3\times x^{2} + 7 = 12x^2 + 7$