Calculus Of One Real Variable By Pheng Kim Ving
Chapter 3: Rules Of Differentiation Section 3.2: Differentiation Of Products And Quotients

 

3.2
Differentiation Of Products And Quotients

 

 

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1. Differentiation Of Products Of Functions

 

Recall from algebra that the product of functions f and g is the function denoted by fg and defined by ( fg)(x) =
f(x)g(x). The value of the product fg at x is the product of the values of f and g at x. We want to find the derivative of
fg. Let's consider an example. Let f(x) = 2x and g(x) = x3. So ( fg)(x) = (2x)(x3) = 2x4. We have f '(x) = 2, g'(x) =
3x2, and ( fg)'(x) = 8x3. Now, f '(x)g'(x) = (2)(3x2) = 6x2. Thus, ( fg)'(x) is not equal to f '(x)g'(x).

 

We'll see in the theorem below that the derivative ( fg)' of fg is f 'g + fg', not f 'g'! The derivative of the product is not
the product of the derivatives!

 

Theorem 1.1 The Product Rule

 

If f and g are differentiable, then fg is also differentiable, and:

 

( fg)' = f 'g + fg'.

 

 

Proof
Let x be an arbitrary point where both f and g are differentiable. Then:

 

 


EOP

 

Recall from Section 2.4 Theorem 1.1 that if a function is differentiable at a point, then it's continuous there. Note also that

 

The General Product Rule

 

The product rule can be extended to more than two factors. If f, g, and h are differentiable, then fgh is also differentiable,
and:

 

( fgh)' = f '( gh) + f( gh)' = f 'gh + f( g'h + gh' ) = f 'gh + fg'h + fgh'.

 

In general, if f1, f2, ..., fn are differentiable, then f1 f2 ...fn is also differentiable, and:

 

( f1 f2...fn)' = f1'f2...fn + f1 f2'...fn + ... + f1 f2...fn'.

 

Example 1.1

 

Find the derivative of y = (x2 + 1)(x3 2).

 

Solution
y' = 2x(x3 2) + (x2 + 1)(3x2) = 2x4 4x + 3x4 + 3x2 = 5x4 + 3x2 4x.
EOS

 

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2. Differentiation Of The Square Root Function

 

Corollary 2.1

 

We have:

 

 

for all x > 0.

 

 

Proof
Using the product rule we get:

 

EOP

 

Example 2.1

 

 

Solution

EOS

 

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3. Differentiation Of Reciprocals Of Functions

 

 

Theorem 3.1 The Reciprocal Rule

 

 

 

Proof

 

 

EOP

 

At any point x where f(x) = 0, f is differentiable, but 1/f isn't, because 1/f isn't defined there.

 

A Special Case

 

A special case is the derivative of 1/x. Since (d/dx) x = 1 we have:

 

 

 

 

Example 3.1

 

Find:

 

 

Solution

EOS

 

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4. The Power Rule The Derivative Of xn

 

In Section 3.1 Theorem 4.1 we see that (d/dx) xn = nxn1 for all positive integer n. We now extend this formula to all
integers.

 

Corollary 4.1 The Power Rule

 

We have:

 

 

for all integer n.

 

 

Proof

EOP

 

We'll see in Section 6.3 Eq. [4.1] that (d/dx) xa = axa1 for any real number a. That's called the general power rule.

 

Example 4.1

 

Find the derivative of f(t) = 2t3 + 4t3.

 

Solution 1
f '(t) = 6t2 12t4.

EOS

 

In this solution we use the power rule on both terms 2t3 and 4t3 of f(t). The reciprocal rule can also be utilized on 4t3,
as done in Solution 2 below.

 

Solution 2

EOS

 

We see that the power rule is much simpler than the reciprocal rule.

 

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5. Differentiation Of Quotients Of Functions

 

 

Theorem 5.1 The Quotient Rule

 

 

 

Proof

EOP

 

Example 5.1

 

Evaluate:

 

 

Solution

EOS

 

Remark that we substitute the value x = 2 without first simplifying. It's simpler to do numerical calculations than to handle
algebraic symbols like the letter x.

 

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Problems & Solutions

 

1. Differentiate the following functions.

 

 

Solution

 

 

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2. Evaluate:

 

 

Solution

 

 

Note

 

We evaluate the derivative immediately after it's calculated, before any simplification takes place. That's because it's
easier to simplify an expression with numbers than with algebraic symbols.

 

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3. Find the tangent and normal lines to the curve y = (x + 1)/(x 1) at x = 2.

 

Solution

 

 

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4. Find all points on the curve y = x + (1/x) where the tangent line is horizontal.

 

Solution

 

 

 

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5. Let b be a non-zero constant. Find the line that passes thru the point (0, b) and is tangent to the curve y = 1/x.

 

Solution

 

Suppose the line is tangent to the curve at x = a. Then the slope of the line is:

 

 

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