TBIL-Cal The Product and Quotient Rules (DF4)

The product rule is a powerful tool, but sometimes it isn’t necessary; a more elementary rule may suffice. For which of the following functions can you find the derivative without using the product rule? Select all that apply.

$f (x) = e^{x} \sin x$
$f (x) = \sqrt{x} (x^{3} + 3 x - 3)$
$f (x) = (4) (x^{5})$
$f (x) = x \ln x$

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Activity 2.4.4.

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Find the derivative of the following functions using the product rule.

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(a)

f (x) = (x^{2} + 3 x) \sin x

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(b)

f (x) = e^{x} \cos x

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(c)

f (x) = x^{2} \ln x

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Activity 2.4.5.

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Let

f

and

g

be the functions defined by

f (t) = 2 t^{2}, g (t) = t^{3} + 4 t .

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(a)

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Determine

f^{'} (t)

and

g^{'} (t) .

(You found these previously in Activity 2.4.1.)

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(b)

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Let

Q (t) = \frac{t^{3} + 4 t}{2 t^{2}}

and observe that

Q (t) = \frac{g (t)}{f (t)} .

Rewrite the formula for

Q

by dividing each term in the numerator by the denominator and use rules of exponents to write

Q

as a sum of scalar multiples of power functions. Then, compute

Q^{'} (t)

using the sum and scalar multiple rules.

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(c)

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True or false:

Q^{'} (t) = \frac{g^{'} (t)}{f^{'} (t)} .

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Theorem 2.4.6. Quotient Rule.

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f

and

g

are differentiable functions, then their quotient

Q (x) = \frac{f (x)}{g (x)}

is also a differentiable function for all

x

where

g (x) \neq 0

and

Q^{'} (x) = \frac{f^{'} (x) \cdot g (x) - f (x) \cdot g^{'} (x)}{g (x)^{2}} .

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Activity 2.4.7.

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Just like with the product rule, there are times when we can find the derivative of a quotient using elementary rules rather than the quotient rule. For which of the following functions can you find the derivative without using the quotient rule? Select all that apply.

$f (x) = \frac{6}{x^{3}}$
$f (x) = \frac{2}{\ln x}$
$f (x) = \frac{e^{x}}{\sin x}$
$f (x) = \frac{x^{3} + 3 x}{x}$

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Activity 2.4.8.

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Find the derivative of the following functions using the quotient rule (or, if applicable, an elementary rule).

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(a)

f (x) = \frac{6}{x^{3}}

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(b)

f (x) = \frac{2}{\ln x}

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(c)

f (x) = \frac{e^{x}}{\sin x}

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(d)

f (x) = \frac{x^{3} + 3 x}{x}

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Activity 2.4.9.

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Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (product, quotient, sum and difference, etc.) you are using in your work.

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(a)

f (w) = - \frac{3 w^{2} + 5 w - 2}{\sin (w)}

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(b)

g (t) = \frac{t^{2} + 6 t + 1}{t^{2}}

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(c)

h (t) = - 2 (t^{2} + 3 t + 3) \cos (t)

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Note 2.4.10.

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We have found the derivatives of

\sin x

and

\cos x,

but what about the other trigonometric functions? It turns out that the quotient rule along with some trig identities can help us! (See Khan Academy

KhanAcademy.org

for a reminder of trig identities.)

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Activity 2.4.11.

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Consider the function

f (x) = \tan x,

and remember that

\tan x = \frac{\sin x}{\cos x} .

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(a)

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What is the domain of

f ?

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(b)

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Use the quotient rule to show that one expression for

f^{'} (x)

f^{'} (x) = \frac{(\cos x) (\cos x) + (\sin x) (\sin x)}{(\cos x)^{2}} .

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(c)

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Which trig identity might be useful here to simplify this expression? How can this identity be used to find a simpler form for

f^{'} (x) ?

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(d)

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Recall that

\sec x = \frac{1}{\cos x} .

How can we express

f^{'} (x)

in terms of the secant function?

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(e)

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For what values of

x

f^{'} (x)

defined? How does this domain compare to the domain of

f ?

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Activity 2.4.12.

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Let

g (x) = \cot x,

and recall that

\cot x = \frac{\cos x}{\sin x} .

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(a)

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What is the domain of

g (x) ?

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(b)

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Use the quotient rule to develop a formula for

g^{'} (x)

that is expressed completely in terms of

\sin x

and

\cos x .

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(c)

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Use other relationships among trigonometric functions to write

g^{'} (x)

only in terms of the cosecant function.

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(d)

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What is the domain of

g^{'} (x) ?

How does this domain compare to the domain of

g^{'} (x) ?

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Activity 2.4.13.

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Let

h (x) = \sec x,

and recall that

\sec x = \frac{1}{\cos x} .

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(a)

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What is the domain of

h (x) ?

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(b)

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Use the quotient rule to develop a formula for

h^{'} (x)

that is expressed completely in terms of

\sin x

and

\cos x .

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(c)

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Use other relationships among trigonometric functions to write

h^{'} (x)

only in terms of the the tangent and secant functions.

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(d)

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What is the domain of

h^{'} (x) ?

How does this domain compare to the domain of

h^{'} (x) ?

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Activity 2.4.14.

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Let

p (x) = \csc x,

and recall that

\csc x = \frac{1}{\sin x} .

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(a)

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What is the domain of

p (x) ?

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(b)

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Use the quotient rule to develop a formula for

p^{'} (x)

that is expressed completely in terms of

\sin x

and

\cos x .

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(c)

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Use other relationships among trigonometric functions to write

h^{'} (x)

only in terms of the the cotangent and cosecant functions.

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(d)

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What is the domain of

p^{'} (x) ?

How does this domain compare to the domain of

p^{'} (x) ?

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Theorem 2.4.15.

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We can now summarize the derivatives of all six trigonometric functions.

$\frac{d}{d x} \sin x = \cos x$
$\frac{d}{d x} \cos x = - \sin x$
$\frac{d}{d x} \tan x = (\sec x)^{2}$
$\frac{d}{d x} \cot x = - (\csc x)^{2}$
$\frac{d}{d x} \sec x = \sec x \tan x$
$\frac{d}{d x} \csc x = - \csc x \cot x$

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Activity 2.4.16.

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Consider the functions

f (x) = 3 \cos (x), g (x) = x^{2} + 3 e^{x}

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and the function

h (x)

for which a table of values is given.

\begin{array}{cccc} x & - 1 & 0 & 2 \\ h (x) & - 4 & - 1 & 3 \\ h^{'} (x) & 0 & - 1 & 1 \end{array}

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In answering the following questions, be sure to explicitly denote which derivative rules (product, quotient, sum/difference, etc.) you are using in your work.

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