5. Derivatives of Polynomials

by M. Bourne

The good news is we can find the derivatives of polynomial expressions without using the delta method that we met in The Derivative from First Principles.

Isaac Newton and Gottfried Leibniz obtained these rules in the early 18th century. They follow from the "first principles" approach to differentiating, and make life much easier for us.

Common derivatives

a. Derivative of a Constant

`(dc)/(dx)=0`

This is basic. In English, it means that if a quantity has a constant value, then the rate of change is zero.

Example a

`d/dx(6)=0`

b. Derivative of n-th power of x

`d/(dx)x^n=nx^(n-1)`

This follows from the delta method.

Example b

`d/(dx)x^5=5x^4`

c. Derivative of Constant product

`d/(dx)(cy)=c(d)/(dx)(y)=c(dy)/dx`

Here, y is some function of x. It means that if we are finding the derivative of a constant times that function, it is the same as finding the derivative of the function first, then multiplying by the constant.

Example c

If `y=x^7`, then `dy/dx=d/dx(x^7)=7x^6`.

Applying the new rule (c), we have:

`d/(dx)(3y)=d/(dx)(3x^7)`

`=3(d)/(dx)(x^7)`

`=3(dy)/dx`

`=(3)(7x^6)`

`=21x^6`

d. Derivative of a sum

`(d(u+v))/(dx)=(du)/(dx)+(dv)/(dx)`

Here, u and v are functions of x. The derivative of the sum is simply equal to the derivative of the first plus derivative of the second. It does not work the same for the derivative of the product of two functions, that we meet in the next section.

Example d

If `u=x^2` and `v=x^9`, then:

`(d(u+v))/(dx)=d/dx(x^2)+d/dx(x^9)`

`=2x+9x^8`

Derivatives Summary

Constant: `(dc)/(dx)=0`
n-th power of x: `d/(dx)x^n=nx^(n-1`
Constant product: `d/(dx)(cy)` `=c(d)/(dx)(y)=c(dy)/dx`
Sum: `(d(u+v))/(dx)=(du)/(dx)+(dv)/(dx)`

Continues below

Further Examples

Example 1

Find the derivative of y = −7x6

Example 2

Find the derivative of y = 3x5 − 1

Example 3

Find the derivative of

`y=13x^4-6x^3-x-1`

Example 4

Find the derivative of

`y=-1/4x^8+1/2x^4-3^2`

Example 5

Evaluate the derivative of

`y=x^4-9x^2-5x`

at the point `(3,15)`.

Example 6

Find the derivative of the function

`y=x^(1//4)-2/x`

Exercise

Find the equation of the tangent to the curve `y = 3x − x^3` at `x = 2`.