Chap 23 Review

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Exercise 1 What’s the derivative of $x^3$ with respect to $x$? Solve this by writing $x^3$ as $x\cdot x^2$ and applying the product rule.

Answer

Since we already know ${\partial }_{x}x$ (it is 1) and ${\partial }_x{x}^2$ (it is $2x$) let’s apply the product rule to find ${\partial }_x{x}^3$: $${\partial }_x[x\times x^2]=[{\partial }_{x}x]\times x^2+x\times [{\partial }_x{x}^2]=1\times x^2+x\times 2x=3x^2$$

Exercise 33 Recognizing $e^{2x}$ as $e^x\times e^x$, apply the product rule.

Answer

The functions being multiplied are identical: $f(x)\equiv e^x$ and $g(x)\equiv e^x$.

Naturally their derivatives are also identical, and since $e^x$ is involved they will both be $e^x$:

$$f^{\prime }(x)=e^x\text{and}{g}^{\prime }(x)=e^x.$$

The formula for the product rule involves $f^{\prime }()g()$ and $g^{\prime }()f()$. Each of these are $e^x{e}^x$ which is the same as $e^{2x}$. The sum $f^{\prime }()g()+g^{\prime }()f()$ is therefore $2e^{2x}$.

Exercise 2 Use the chain rule to find the derivative ${\partial }_x{e}^{2x}$.

Hint: The first step is to identify the interior $f()$ and the exterior $g()$ functions. Then differentiate each to get $f^{\prime }()$ and $g()$ and apply the formula.

Answer

$g(x)\equiv 2x$ is the interior function in $e^{2x}$ and $f(x)\equiv \exp (x)$ is the exterior function. Thus $${\partial }_x{e}^{2x}=f^{\prime }(g(x))g^{\prime }(x)=\exp (g(x))2=2e^{2x}.$$

Exercise 3 Which of the derivative rules should you use to find $${\partial }_t{e}^{t^2}?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-1

Exercise 4 Which of the derivative rules should you use to find $${\partial }_t{e}^{x^2}?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-2

Exercise 5 Which of the derivative rules should you use to find $${\partial }_t{e}^t\sin (t)?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-3

Exercise 6 Which of the derivative rules should you use to find $${\partial }_t{e}^t\sin (x)?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-4

Exercise 7 Which of the derivative rules should you use to find $${\partial }_t\ln (t)?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-5

Exercise 8 Which of the derivative rules should you use to find $${\partial }_{t}t{e}^{-t}?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-6

Exercise 9 Which of the derivative rules should you use to find $${\partial }_{x}37x^5?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-7

Exercise 10 Which of the derivative rules should you use to find $${\partial }_{x}19?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-8

Exercise 11 Which of the derivative rules should you use to find $${\partial }_{x}15x^2-3x+7\ln (x)?$$

The constant multiplier rule

The linear combination rule

The product rule

The chain rule

No rule needed, it is so basic.

question id: drill-Deriv-rules-9

Exercise 12 What is ${\partial }_{x}15x^2-3x+7\ln (x)$?

$15x-3+7/x$

$30x-3+7/x$

$30x-3x+7/x$

$30x-3-7/x$

question id: drill-Deriv-rules-10

Exercise 13 What is ${\partial }_t[e^k+\ln (e^2)-t]$?

-1       0       $e^k+1/e$       $k{e}^k+2/e-t$      

question id: drill-Deriv-rules-11

Exercise 14 What is ${\partial }_x\ln (x)/x^2$? (Hint: You can write the function in a simpler way.)

$-2x^{-1}\ln (x)$       $-2x^{-3}\ln (x)$       $x^{-3}(1-2\ln (x))$       $-2x^{-3}(1/x-1)$      

question id: drill-Deriv-rules-12

Exercise 15 Which of these is ${\partial }_t(ln(6)+t^4-e^t)$?

$\frac{1}{6}+4t^3-e^t$

$\frac{1}{6}+4t^3-e^{-t}$

$4t^3-e^{-t}$

$4t^3-e^t$

question id: drill-Deriv-rules-12a

Exercise 16 Which of these is ${\partial }_u(\frac{1}{u^6}-{\pi }^3+4u^3+e)$?

$-6u^{-7}-3{\pi }^2+4u^3$

$-6u^{-5}-3{\pi }^2+12u^2+\frac{1}{e}$

$-6u^{-7}+12u^2$

$-6u^{-5}+12u^2+\frac{1}{e}$

question id: drill-Deriv-rules-12b

Exercise 17 Which of these is ${\partial }_v(\sqrt[4]{v^7}+e^7-4-\frac{3v^6}{v^2})=$

$\frac{7}{4}\frac{1}{v^4}+7e^6-\frac{18v^5}{2v}$

$\frac{7}{4}{v}^{\frac{3}{4}}-12v^3$

$\frac{4}{7}{v}^{\frac{-3}{7}}-\frac{18v^5}{2v}$

$\frac{7}{4}{v}^{\frac{3}{4}}+e^7-12v^3$

question id: drill-Deriv-rules-12c

Exercise 18 What is ${\partial }_t(4\sin (2\pi t)-5)$?

$4\cos (2\pi t)-5$       $4\pi \cos (2\pi t)$       $8\pi \cos (2\pi t)$       $8\cos (2\pi t)$      

question id: drill-Deriv-rules-13

Exercise 19 What is ${\partial }_t(7+8t^2+3t^4)$?

$16t+12t^3$       $8t+4t^3$       $16t^2+9t^3$       $4t+12t^2$      

question id: drill-Deriv-rules-14

Exercise 20 The derivative ${\partial }_x\text{dnorm}(x)=-x\text{dnorm}(x)$. What is $${\partial }_x\text{dnorm}\left(\frac{x^2}{4}\right)?$$

$-\frac{x^3}{8}\text{dnorm}\left(\frac{x^2}{4}\right)$

$-\frac{x}{2}\text{dnorm}\left(\frac{x^2}{4}\right)$

$-\frac{x}{8}\text{dnorm}\left(\frac{x^2}{4}\right)$

$-\frac{x^2}{2}\text{dnorm}\left(\frac{x^2}{4}\right)$

question id: drill-Deriv-rules-15

Exercise 21 What is ${\partial }_t(6t-3t^2+2t^4)$?

$6-6t+8t^3$       $6-3t+6t^3$       $-3t+6t^3$       $6-3t+8t^2$      

question id: drill-Deriv-rules-16

Exercise 22 What is ${\partial }_t\ln (t^2+1)$?

$\frac{2t}{t^2+1}$       $1/t^2+1$       $1/2t$       $2t\ln (t^2+1)$      

question id: drill-Deriv-rules-17

Exercise 23 For the function $$g(t)\equiv \sin (\frac{2\pi }{P}(t-t_0))$$ is the interior function linear?

Yes       No      

question id: drill-M08-1

Exercise 24 For the function $$g(P)\equiv \sin (\frac{2\pi }{P}(t-t_0))$$ is the interior function linear?

Yes       No      

question id: drill-M08-2

Exercise 25 For the function $$h(u)\equiv \ln (a^{2}u-\sqrt{b})$$ is the interior function linear?

Yes       No      

question id: drill-M08-3

Exercise 26 For the function $f(w)\equiv e^{kw}$, is the interior function linear?

Yes       No      

question id: drill-M08-4

Exercise 27 Saying “the interior function is linear” is not an entirely complete statement. A full statement is “the interior function is linear in terms of the input $x$” or “in terms of the input $u$” or whatever name we choose to use for the input.
Is the expression $Vx+U$ linear in terms of $U$?

Yes       No      

question id: drill-M08-5

Exercise 28 Saying “the interior function is linear” is not an entirely complete statement. The full statement is “the interior function is linear in terms of the input $x$” or “in terms of the input $u$” or whatever name we choose to use for the input.

Is the expression $V{x}^2+U$ linear in terms of $U$?

Yes       No      

question id: drill-M08-6

Exercise 29 Here are several functions that are related by differentiation and integration:

1.   $\frac{1}{a}{e}^{ax+b}$
2.   $a^2{e}^{ax+b}$
3.   $\frac{1}{a^2}{e}^{ax+b}$
4.   $e^{ax+b}$
5.   $a{e}^{ax+b}$

Put these functions in order that the derivative of each function preceeds the anti-derivative.

The order is b-e-d-a-c

The order is b-d-e-c-a

The order is b-d-c-e-a

The order is e-b-d-c-a

The order is e-b-d-a-c

question id: naCVxW

Exercise 30 Here are several functions that are related by differentiation and integration:

1.   $-\cos (x)$
2.   $\cos (x)$
3.   $\sin (x)$
4.   $-\sin (x)$

Put these functions in order that the derivative of each function preceeds the anti-derivative.

a-b-d-c

b-d-c-a

d-b-a-c

a-d-b-c

b-a-d-c

question id: chain-of-differentiation-2

Exercise 31 Saying “the interior function is linear” is not an entirely complete statement. A full statement is “the interior function is linear in terms of the input $x$” or “in terms of the input $u$” or whatever name we choose to use for the input.
Is the expression $V{x}^2+U$ linear in terms of $X$?

Yes       No      

question id: drill-M08-7

Exercise 32  

In draft

Make this an exercise in three parts: do the derivative using power-law rules, then using multiplication, then using composition.

There is one family of functions for which function composition accomplishes same thing as multiplying functions: the power-law family.

Consider, for instance, the function $h(x)\equiv {\left[3x\right]}^4$. Let’s let $g(x)\equiv 3x$ and $f(y)\equiv y^4$. With these definitions, $h(x)=f(g(x))$.

Recognizing that ${\partial }_{y}f(y)=4y^3$ and ${\partial }_{x}g(x)=3$, the chain rule gives $${\partial }_{x}h(x)=\underset{f^{\prime }(g(x))}{\underbrace{4g(x{)}^3}}\times \underset{g^{\prime }(x)}{\underbrace{3}}=\underset{f^{\prime }(g(x))}{\underbrace{4(3x{)}^3}}\times 3=4\cdot 3^4\times x^3=324x^3$$ Another way to look at the same function is $g(x)$ multiplied by itself 3 times: $$h(x)=g(x)\cdot g(x)\cdot g(x)\cdot g(x)$$ This is a product of 4 terms. Applying the product rule gives $$\begin{array}{rcl}{\partial }_{x}h(x)& =& g^{\prime }(x)\cdot g(x)\cdot g(x)\cdot g(x)+\\ & & g(x)\cdot {g(x)}^{\prime }\cdot g(x)\cdot g(x)+\\ & & g(x)\cdot g(x)\cdot g(x{)}^{\prime }\cdot g(x)+\\ & & g(x)\cdot g(x)\cdot g(x)\cdot g^{\prime }(x)\end{array}$$ Since multiplication is commutative, all four terms are the same, each being $3^4{x}^3$. The sum of all four is therefore $4\times 3^4{x}^3=324x^3$.

These are two long-winded ways of getting to the result. For most people, differentiating power-law functions algebraically is simplified by using the rules of exponentiation rather than the product or chain rule. Here, $$h(x)\equiv {\left[3x\right]}^4=3^4{x}^4$$so ${\partial }_{x}h(x)$ is easily handled as a scalar ($3^4$) times a function $x^4$. Consequently, applying the rule for differentiating power laws, $${\partial }_{x}h(x)=3^4\times {\partial }_x{x}^4=3^4\times 4x^3=324x^3$$ As another example, take $h(x)\equiv \sqrt[4]{x^3}$. This is, of course, the composition $f(g(x))$ where $f(y)\equiv y^{1/4}$ and $g(x)\equiv x^3$. Applying the chain rule to find ${\partial }_{x}h(x)$ will work (of course!), but is more work than applying the rules of exponentiation followed by a simple power-law differentiation. $$h(x)=\sqrt[4]{x^3}=x^{3/4}\text{so}{\partial }_{x}h(x)=\frac{3}{4}{x}^{(3/4-1)}=\frac{3}{4}{x}^{-1/4}$$

Exercise 34 For the function $$g(t)\equiv \sin (\frac{2\pi }{P}(t-t_0))$$ is the interior function linear?

Yes       No      

question id: interior1

Exercise 35 For the function $$g(P)\equiv \sin (\frac{2\pi }{P}(t-t_0))$$ is the interior function linear?

Yes       No      

question id: interior2

Exercise 36 For the function $$h(u)\equiv \ln (a^{2}u-\sqrt{b})$$ is the interior function linear?

Yes       No      

question id: interior3

Exercise 37 For the function $f(w)\equiv e^{kw}$, is the interior function linear?

Yes       No      

question id: interior4

Exercise 38 Saying “the interior function is linear” is not an entirely complete statement. A full statement is “the interior function is linear in terms of the input $x$” or “in terms of the input $u$” or whatever name we choose to use for the input.
Is the expression $Vx+U$ linear in terms of $U$?

Yes       No      

question id: interior5

Exercise 39 Saying “the interior function is linear” is not an entirely complete statement. A full statement is “the interior function is linear in terms of the input $x$” or “in terms of the input $u$” or whatever name we choose to use for the input.
Is the expression $V{x}^2+U$ linear in terms of $U$?

Yes       No      

question id: interior6

Exercise 40 Saying “the interior function is linear” is not an entirely complete statement. A full statement is “the interior function is linear in terms of the input $x$” or “in terms of the input $u$” or whatever name we choose to use for the input.
Is the expression $V{x}^2+U$ linear in terms of $X$?

Yes       No      

question id: interior7

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