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
Chap 23 Review
\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]
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 \(\partial_x x^3\): \[\partial_x [\color{magenta}x \times \color{brown}{x^2}] = \color{magenta}{[\partial_x x]} \times \color{brown}{x^2} \ + \color{magenta}x \times \color{brown}{[\partial_x x^2]} =\color{magenta}1\times \color{brown}{x^2} + \color{magenta}x \times \color{brown}{2x} = 3 x^2\]
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 \(e^x\):
\[f'(x) = e^x\ \ \ \ \text{and} \ \ \ \ g'(x) = e^x \ .\]
The formula for the product rule involves \(f'() g()\) and \(g'() f()\). Each of these are \(e^x\ e^x\) which is the same as \(e^{2x}\). The sum \(f'() g() + g'() f()\) is therefore \(2 e^{2x}\).
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 \(f'()\) 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 \[\partial_x e^{2x} = f'\left(\strut g(x)\right) g'(x) = \exp\left(\strut g(x)\right) 2 = 2 e^{2x}\ .\]
Exercise 3 Which of the derivative rules should you use to find \[\partial_t e^{t^2}\ ?\]
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 \[\partial_x\ 37 x^5\ ?\]
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 \[\partial_x\ 15 x^2 - 3 x + 7 \ln(x)\ ?\]
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 \(\partial_x\ 15 x^2 - 3 x + 7 \ln(x)\)?
\(15 x - 3 + 7/x\)
\(30 x - 3 + 7/x\)
\(30 x - 3x + 7/x\)
\(30 x - 3 - 7/x\)
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.)
\(-2 x^{-1} \ln(x)\) \(-2 x^{-3} \ln(x)\) \(x^{-3} \left(1 - 2 \ln(x)\right)\) \(-2 x^{-3} \left(1/x - 1\right)\)
question id: drill-Deriv-rules-12
Exercise 15 Which of these is \(\partial_t \left(ln(6)+t^4-e^t\right)\)?
\(\frac{1}{6}+4t^3-e^t\)
\(\frac{1}{6}+4t^3-e^{-t}\)
\(4t^3-e^{-t}\)
\(4t^3-e^t\)
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 \(\partial_{t} \left(4 \sin(2\pi t) - 5\right)\)?
\(4 \cos(2\pi t) - 5\) \(4 \pi \cos(2 \pi t)\) \(8 \pi \cos(2 \pi t)\) \(8 \cos(2 \pi t)\)
question id: drill-Deriv-rules-13
Exercise 19 What is \(\partial_{t} \left(7 + 8 t^2 + 3 t^4\right)\)?
\(16 t + 12 t^3\) \(8 t + 4 t^3\) \(16 t^2 + 9 t^3\) \(4 t + 12 t^2\)
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 \(\partial_{t} \left(6 t - 3 t^2 + 2 t^4\right)\)?
\(6 - 6 t + 8 t^3\) \(6 - 3 t + 6 t^3\) \(-3 t + 6 t^3\) \(6 - 3 t + 8 t^2\)
question id: drill-Deriv-rules-16
Exercise 22 What is \(\partial_t \ln(t^2 + 1)\)?
\(\frac{2t}{t^2+1}\) \(1/{t^2 + 1}\) \(1/2t\) \(2 t \ln(t^2 + 1)\)
question id: drill-Deriv-rules-17
Exercise 23 For the function \[g(t) \equiv \sin\left(\frac{2 \pi}{P} (t - t_0)\right)\] is the interior function linear?
Yes No
question id: drill-M08-1
Exercise 24 For the function \[g(P) \equiv \sin\left(\frac{2 \pi}{P} (t - t_0)\right)\] 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 \(V x + U\) 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:
- \(\frac{1}{a} e^{ax + b}\)
- \(a^2 e^{a x + b}\)
- \(\frac{1}{a^2} e^{ax + b}\)
- \(e^{ax + b}\)
- \(a e^{ax + b}\)
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:
- \(-\cos(x)\)
- \(\cos(x)\)
- \(\sin(x)\)
- \(-\sin(x)\)
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
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 \(\partial_y f(y) = 4 y^3\) and \(\partial_x g(x) = 3\), the chain rule gives \[\partial_x h(x) = \underbrace{4 g(x)^3}_{f'(g(x))} \times \underbrace{3}_{g'(x)} = \underbrace{4 (3 x)^3}_{f'(g(x))} \times 3 = 4\cdot 3^4 \times x^3 = 324\ x^3\] 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 \[\begin{eqnarray} \partial_x h(x) &=& \color{blue}{g'(x)}\cdot g(x)\cdot g(x) \cdot g(x) +\\ &\ & g(x)\cdot \color{blue}{g(x)}'\cdot g(x) \cdot g(x) +\\ &\ & g(x)\cdot g(x)\cdot \color{blue}{g(x)'} \cdot g(x) +\\ &\ & g(x)\cdot g(x)\cdot g(x) \cdot \color{blue}{g'(x)} \end{eqnarray}\] Since multiplication is commutative, all four terms are the same, each being \(3^4 x^3\). The sum of all four is therefore \(4 \times 3^4 x^3 = 324 x^3\).
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, \[\partial_x h(x) = 3^4 \times \partial_x x^4 = 3^4 \times 4 x^3 = 324 x^3\] As another example, take \(h(x) \equiv \sqrt[4]{\strut x^3}\). 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. \[h(x) = \sqrt[4]{\strut x^3} = x^{3/4}\ \ \text{so}\ \ \partial_x h(x) = \frac{3}{4} x^{(3/4 - 1)} = \frac{3}{4} x^{-1/4}\]
Exercise 34 For the function \[g(t) \equiv \sin\left(\frac{2 \pi}{P} (t - t_0)\right)\] is the interior function linear?
Yes No
question id: interior1
Exercise 35 For the function \[g(P) \equiv \sin\left(\frac{2 \pi}{P} (t - t_0)\right)\] 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 \(V x + U\) 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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