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.
|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.|
|n-th power of x:||`d/(dx)x^n=nx^(n-1`||This follows from the delta method.|
|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.|
|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 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.|
1. Find the derivative of y = −7x6
2. Find the derivative of y = 3x5 − 1
3. Find the derivative of
4. Find the derivative of
5. Evaluate the derivative of
at the point `(3,15)`.
6. Find the derivative of the function
Find the equation of the tangent to the curve `y = 3x − x^3` at `x = 2`.
Didn't find what you are looking for on this page? Try search:
Online Algebra Solver
This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)
Go to: Online algebra solver
Ready for a break?
Play a math game.
(Well, not really a math game, but each game was made using math...)
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
Short URL for this Page
Save typing! You can use this URL to reach this page:
Calculus Lessons on DVD
Easy to understand calculus lessons on DVD. See samples before you commit.
More info: Calculus videos