Derivatives of Polynomial Functions
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
[Tex]\frac{\partial (x^{n})}{\partial x} = n\times x^{n-1}[/Tex]
(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
[Tex]=> \frac{\partial y}{\partial x} = 3\times x^{3-1} = 3x^2[/Tex]
Some basic rules of derivative
[Tex]\frac{\partial y}{\partial x} = c\frac{\partial (f(x))}{\partial x}[/Tex]
[Tex]\frac{\partial y}{\partial x} = 0[/Tex]
- [Tex]If \ y= f_{1}(x)\pm f_{1}(x)[/Tex]
[Tex]\frac{\partial y}{\partial x} = \frac{\partial (f_{1}(x))}{\partial x}\pm \frac{\partial (f_{1}(x))}{\partial x}\\[/Tex]
Example 1: Find the derivative of 4x3 + 7x?
Solution:
Let y = 4x3 + 7x
[Tex]\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[/Tex]
Example 2: Find the derivative of 3x2 – 7?
Solution:
Let y = 3x2 – 7
[Tex]\frac{\partial y}{\partial x}=6x[/Tex]
Some more examples on derivative of polynomials
Example 1: Find the derivative of [Tex]\frac{1}{x^{7}}[/Tex]?
Solution:
[Tex]Let \ y=\frac{1}{x^{7}}\\[/Tex]
This can be written as
y = x−7
[Tex]\frac{\partial y}{\partial x} = (-7)\times x^{-8}[/Tex]
Example 2: Find the derivative of 7x5 + x3 − x?
Solution:
Let y = 7x5 + x3 − x
[Tex]\frac{\partial y}{\partial x}=35x^{4}+3x^{2}-1[/Tex]
Example 3: Find the derivative of (x + 5)2 + 6x3 − 4?
Solution:
Let y = (x + 5)2 + 6x3 − 4
[Tex]\frac{\partial y}{\partial x} = 2(x+5)+18x^{2}[/Tex]
Example 4: Find the derivative of 6x3 + (6x + 5)2 − 8x?
Solution:
Let y = 6x3 + (6x + 5)2 − 8x
[Tex]\frac{\partial y}{\partial x} = 18x^{2}+2(6x+5)(6)-8\\ \frac{\partial y}{\partial x} =18x^{2}+12(6x+5)-8[/Tex]
Example 5: Find the derivative of [Tex]\frac{1}{(2x+8)^{7}}[/Tex]?
Solution:
[Tex]Let \ y=\frac{1}{(2x+8)^{7}}\\ y=(2x+8)^{-7}\\ \frac{\partial y}{\partial x}=(-7)(2x+8)^{-8}(2)\\ \frac{\partial y}{\partial x}=(-14)(2x+8)^{-8}[/Tex]
Last Updated :
26 Feb, 2024
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