Chapter Contents

• Differentiation
• 1. Limits and Differentiation
• 2. The Slope of a Tangent to a Curve (Numerical)
• 3. The Derivative from First Principles
• 4. Derivative as an Instantaneous Rate of Change
• 5. Derivatives of Polynomials
• 6. Derivatives of Products and Quotients
• Derivative interactive graph
• Differentiation Java applet
• Derivative of a Cubic - Exploration using GeoGebra
• 7. Differentiating Powers of a Function
• 8. Differentiation of Implicit Functions
• 9. Higher Derivatives
• 10. Partial Derivatives
• Differentiation Problem Solver

• Comments, Questions?

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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 18^th 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.

Examples

1. Find the derivative of y = −7x⁶

Answer

2. Find the derivative of y = 3x⁵ − 1

Answer

3. Find the derivative of

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

Answer

4. Find the derivative of

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

Answer

5. Evaluate the derivative of

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

at the point `(3,15)`.

Answer

6. Find the derivative of the function

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

Answer

Exercise

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

Answer