6. Derivatives of Products and Quotients

by M. Bourne


PRODUCT RULE

If u and v are two functions of x, then the derivative of the product uv is given by...

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Later in this section:
Quotient Rule

math formula

In words, this can be remembered as:

"The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first."

Example 1:

If we have a product like

y = (2x2 + 6x)(2x3 + 5x2)

we can find the derivative without multiplying out the expression on the right.


Answer


LiveMath does things in a slightly different order...

LIVEMath

 

Example 2:

Find the derivative of math formula


Answer


NOTE:

We can write the product rule in many different ways:

math formula

OR

math formula… etc.

 


QUOTIENT RULE

(A quotient is just a fraction.)

If u and v are two functions of x, then the derivative of the quotient u/v is given by...

math formula

In words, this can be remembered as:

"The derivative of a quotient equals bottom times derivative of top minus top times derivative of the bottom, divided by bottom squared."

Example 1:

We wish to find the derivative of the expression:

math formula


Answer


Play with this example on the Differentiation Java applet page.

Example 2:

Find math formula if math formula.


Answer

Play with this example on the Differentiation Java applet page.

Let's also see this in LiveMath.

LIVEMath


NOTE:

You will also see the quotient rule written as:

math formula

OR

math formula etc.



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