**The Product Rule**

When a function consists of parts that are added together, it is easy to take its derivative: Simply take the derivative of each part and add them together. We are inclined to try the same trick when the parts are multiplied together, but it does not work.

For example, we know that and .

The derivative of their product is .

This shows that the derivative of a product is *not* the product of the derivatives:

.

Instead, we take the derivative of each part, multiply by the other part *left alone*, and add the results together:

This time, *we did* get the correct answer.

*The Product Rule*

*The Product Rule*

*The Product Rule can be stated "the derivative of the first times the second, plus the derivative of the second times the first." It can be proven directly from the limit definition of the derivative, but only with a few tricks and a lot of algebra. The Product Rule is given as follows:*

**Example 1**

Differentiate *y* = *x*^{3}sin(*x*).

**Solution 1**

Here, the "first part" is *x*^{3} and the "second part" is sin(*x*). Thus, by using the Product Rule, = 3*x*^{2}sin(*x*) + cos(*x*) · *x*^{3} . This could be simplified as , but that's not really all that's necessary.

**Example 2**

Differentiate f(*x*) = In(*x*) · cos(*x*).

**Solution 2**

Thus, the derivative is:

** Example 3**

Differentiate *g*(*x*) = 5*x*^{3} · *e*^{x}.

**Solution 3**

= 35*x*^{6} · *e*^{x} + *e*^{x} · 5*x*^{3} =5*x*^{6}*e*^{x} (7 + *x*)

Using the product rule with *e*^{x} can be a little bit confusing because there is no difference between the derivative of *e*^{x} and *e*^{x} "left alone." Still, if you write everything out, the correct answer should fall into place, even if it looks weird.

**Example 4**

Differentiate *y* = *t*^{2}ln(*t*).

**Solution 4**

=

= 2*t* · ln(*t*) + *t*

= *t*(2 + In(*t*))

**Example 5**

Differentiate *y* = *x*^{5}sin(*x*)cos(*x*).

**Solution 5**

We'll use the Product Rule with *x*^{5} as the first part and sin(*x*)cos(*x*) as the second part. However, in taking the derivative of sin(*x*)cos(*x*), we'll have to use the Product Rule a second time. It might get messy, but it will work if everything is written down carefully.

[cos(*x*) · cos(*x*) – sin(*x*) · sin(*x*)] · *x*^{5}

**The Quotient Rule**

The Quotient Rule for functions where the parts are divided is even more complicated than the Product Rule. The Quotient Rule can be stated:

Just as with the Product Rule, each part is differentiated and multiplied by the other part. Here, however, they are subtracted, so it matters which one is differentiated first. It is important to start with the derivative of the top.

*The Quotient Rule*

*The Quotient Rule*

**Example 1**

Differentiate

**Solution 1**

Here, the top part is *x*^{5} – 3*x*^{2} + 1 and the bottom part is cos(*x*). Therefore, by the Quotient Rule:

**Example 2**

Differentiate

**Solution 2**

**Example 3**

Differentiate

**Solution 3**

Here, the Product Rule is necessary to differentiate the top.

**Example 4**

Differentiate

**Solution 4**

Some people remember the Quotient Rule as just so they can say "HI d'HO" and "HO HO;" but they're silly.

**Derivatives of Trigonometric Functions**

With the Quotient Rule, we can find the derivatives of all of the rest of the trigonometric functions.

**Example 1**

Differentiate *y* = tan(*x*).

**Solution 1**

Differentiate with the Quotient Rule.

Simplify.

Use sin^{2}(*x*) + cos^{2}(*x*) = 1.

Use

Thus:

**Example 2**

Differentiate *y* = sec(*x*).

**Solution 2**

Use

Differentiate with the Quotient Rule.

Simplify.

Use and

Thus:

Find practice problems and solutions for these concepts at The Product and Quotient Rules Practice Questions

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