Derivative+Rules

= ﻿Derivative Rules =

The Power Rule
For f(x) = x^n where n is a natural number, then: Example: y = 7x^4. Find y'.

y' = 4*7x^3. Note: This rule will be used in almost every derivative done. Learn it.

The Constent Multiple Rule
Example: y = 7x^2. Find y'.

7 is the constant multiple. Take the derivative of x^2 and simply multiply it by 7.

The Sum Rule
Example: y = 6x^3 + 8x^4. Find y'.

Treat each part seperately: 6x^3 and 8x^4. Take the derivative of each part seperately: 18x^2 and 32x^3. Then because of the Sum Rule, add them together to get: 18x^2 + 32x^3.

The Subtraction Rule
Example: y = 5x^4 - 3x^3. Find y'.

Treat each part seperately: 5x^4 and 3x^3. Take the derivative of each part seperately: 20x^3 and 9x^2. Then because of the Subtraction Rule, subtract them to get: 20x^3 - 9x^2.

The Product Rule
HELPFUL HINT: one D two plus two D one

Example: y = (7x^5)(3x^2). Find y'.

Treat (7x^5) as part one and treat (3x^2) as part two. Use the HELPFUL HINT to get: (7x^5)(6x) + (3x^2)(35x^4)

The Quotient Rule
(whenever g is nonzero) HELPFUL HINT: Low D high minus high D low, over the square of what's below.

Example: y = (7x^2)/(3x^3). Find y'.

Treat (7x^2) as high and treat (3x^3) as low. Use the HELPFUL HINT to get: [(3x^3)(14x) - (7x^2)(9x^2)]/[(3x^3)^2]

The Chain Rule


Example: y = (8x^3)^2. Find y'.

The chain rule may seem confusing, but an easy way to remember it is to work from the outside to the inside. So (8x^3)^2 is: 2(8x^3)(24x^2).

Derivative of e


Example: y = e^5. Find y'.

Apply the rule for derivatives of e. It is simply: e^5.

Derivatives of Natural Logs


Example: y = ln(7). Find y'.

Apply the rule for natural logs. It is: (1)/(7 ln(e)).

[[image:http://upload.wikimedia.org/math/5/f/0/5f0213d29d4b5f467f1c25330b9a135f.png width="136" height="24" caption=" (sin x)' = cos x ,"]]


The only way to learn these is to memorize them. These derivatives can be applied with any and all of the rules above.

Works Cited

"Differentiation Rules." //Wikipedia, the Free Encyclopedia//. 4 May 2011. Web. 16 May 2011. .