What is a polynomial function and derivatives of polynomial functions

Before learning about the derivatives of polynomial functions, it is good if we know the meaning of polynomials.Polynomial function is a function made up of variables with any powers and constants involving basic algebraic operations like addition, multiplication and subtraction. The generic form a polynomial function is g(x) = a0 + a1x + ... + an-1xn-1 + anxn where a0, a1, ..., an are real coefficients while n is a non-negative integer. Thus it signifies that a polynomial function is made up of one or more power functions. 

Finding the derivatives of polynomial functions

If we know how to find derivatives of power functions, then finding derivatives of polynomial functions becomes really easy. After all a polynomial function is the sum of several power functions. Recalling the equation for a power function, if f(x) = axn, then the derivative f'(x) = anxn-1. This formula can be used as the base for calculating the derivatives of polynomial functions. So taking derivative of generalized polynomial function, we have g'(x) = a1 + a2 * 2x + ... + an-1 * (n – 1)xn-2 + an * nxn-1. Thus taking derivatives of individual power functions inside a polynomial, we get the derivatives of polynomial functions.

Examples for derivatives of polynomial functions

We have seen the general equation for polynomials as well as their derivatives. We will now go through some of the real examples for finding the derivatives to polynomials.

(1) Find the derivative of y = 4x2 + 2x + 1:

Here y is a polynomial function of x. So its derivative can be found by the sum of individual derivatives of all power functions. Derivative of 4x2 will be 8x, that for 2x will be 2 and for any constant will be zero. So dy/dx = 8x + 2.

(2) Find the derivative of a polynomial y = (x-2)2:

Elaborating, y = x2 -4x + 4. Taking the derivative, we get dy/dx = 2x – 4.

(3) Find a polynomial function if its derivative is 4x + 8:

Here dy/dx = 4x + 8. Now for y = axn, we have dy/dx = anxn-1. Using this for 1st power function i.e. 4x, n = 2 and a =2. Also for 2nd power function, n = 1 and a = 8. Hence taking the anti-derivative, we have the polynomial function as y = 2x2 + 8x.

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