Chapter Review
Basic Derivative Formulas:
- \(\dfrac{d}{dx}e^{u}=e^{u}\dfrac{du}{dx}\) (p. 200)
- \(\dfrac{d}{dx}a^{x}=a^{x}\ln a\) \(a>0\) and \(a≠ 1\) (p. 202)
Properties of Derivatives:
- Chain Rule: \(( f\circ g) ^\prime ( x) =f^\prime ( g( x) ) \cdot g^\prime ( x) \) or \(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}\) (p. 198)
- Power Rule for functions: \(\dfrac{d}{dx}[g( x) ]^{n}=n[g( x) ] ^{n-1}g^\prime ( x)\), where \(n\) is an integer (p. 203)
3.2 Implicit Differentiation; Derivatives of the Inverse Trigonometric Functions
Procedure: To differentiate an implicit function (p. 211):
- Assume \(y\) is a differentiable function of \(x\).
- Differentiate both sides of the equation with respect to \(x\).
- Solve the resulting equation for \(y^{\prime}=\dfrac{dy}{dx}\).
Basic Derivative Formulas:
- \(\dfrac{d}{dx}\sin ^{-1}x=\dfrac{1}{\sqrt{1-x^{2}}}\)\(-1<x<1\) (p. 216)
- \(\dfrac{d}{dx}\tan ^{-1}x=\dfrac{1}{1+x^{2}}\) (p. 217)
- \(\dfrac{d}{dx}\sec ^{-1}x=\dfrac{1}{x\sqrt{x^{2}-1}}\)\(\vert x\vert >1\) (p. 218)
Properties of Derivatives:
- Power Rule for rational exponents: \(\dfrac{d}{dx}x^{p/q}=\dfrac{p}{q}\cdot x^{(p/q) -1}\). provided \(x^{p/q}\) and \(x^{p/q-1}\) are defined. (p. 213)
- Power Rule for functions: \(\dfrac{d}{dx}[ u( x) ] ^{r}=r [u( x) ] ^{r-1}u^\prime ( x)\), \(r\) a rational number; provided \(u^{r}\) and \(u^{r-1}\) are defined. (p. 214)
Theorem: The derivative of an inverse function at a number (p. 215)
3.3 Derivatives of Logarithmic Functions
Basic Derivative Formulas:
- \(\dfrac{d}{dx}\log _{a}x=\dfrac{1}{x\ln a}, a>0, a\ne 1\) (p. 222)
- \(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\) (p. 222)
Steps for Using Logarithmic Differentiation (p. 225):
- Step 1 If the function \(y=f( x) \) consists of products, quotients, and powers, take the natural logarithm of each side. Then simplify using properties of logarithms.
- Step 2 Differentiate implicitly, and use \(\dfrac{d}{dx}\ln y=\dfrac{y^\prime }{y}\).
- Step 3 Solve for \(y^\prime \), and replace \(y\) with \(f( x) \).
Theorems:
- Power Rule If \(a\) is a real number, then \(\dfrac{d}{dx} x^{a}=ax^{a-1}\). (p. 226)
- The number \(e\) can be expressed as \(\lim\limits_{h\rightarrow 0}(1+h) ^{1/h}=e\hbox{ or }\lim\limits_{n\rightarrow \infty }\left( 1+\dfrac{1}{n}\right) ^{n}=e\). (p. 227)
3.4 Differentials; Linear Approximations; Newton’s Method
- The differential \(dx\) of \(x\) is defined as \(dx=\Delta x≠ 0,\) where \(\Delta x\) is the change in \(x\). The differential \(dy\) of \(y=f(x)\) is defined as \(dy=f^\prime ( x) dx\). (p. 231)
- A linear approximation \(L( x) \) to a differentiable function \(f\) near \(x=x_{0}\) is given by \(L( x) =f( x_{0}) +f^\prime ( x_{0}) ( x-x_{0})\). (p. 232)
- Newton’s Method for finding the zero of a function. (p. 235)
- Taylor Polynomial \(P_{n}( x)\) for \(f\) at \(x_{0}\): \[ \begin{eqnarray*} P_{n}( x) &=& f( x_{0}) + f^\prime ( x_{0}) (x-x_{0}) + \dfrac{f^{\prime \prime} ( x_{0}) }{2!}( x-x_{0}) ^{2}+\cdots \\ &&+\,\dfrac{f^{( n) }( x_{0}) }{n!}( x-x_{0}) ^{n}\quad (p. 240) \end{eqnarray*} \]
Definitions:
- Hyperbolic sine: \(y=\sinh x=\frac{e^{x}-e^{-x}}{2}\) (p. 243)
- Hyperbolic cosine: \(y=\cosh x=\frac{e^{x}+e^{-x}}{2}\) (p. 243)
Hyperbolic Identities (pp. 244-245):
- \(\tanh x = \frac{\sinh x}{\cosh x}\)
- \(\coth x = \frac{\cosh x}{\sinh x}\)
- \({\rm sech}\, x = \frac{1}{\cosh x}\)
- \({\rm csch}\, x = \frac{1}{\sinh x}\)
- \(\cosh ^{2}x-\sinh ^{2}x=1\)
- \(\tanh ^{2}x+\text{sech}^{2}x=1\)
- \(\coth ^{2}x-\text{csch}^{2}x=1\)
- Sum Formulas: \begin{eqnarray*} \sinh (A+B) &=&\sinh A\cosh B+\cosh A\sinh B\\ \cosh (A+B) &=&\cosh A\cosh B+\sinh A\sinh B \end{eqnarray*}
- Even/odd Properties: \[ \sinh (-A) =-\sinh A\qquad \cosh (-A) =\cosh A \]
Inverse Hyperbolic Functions (p. 247):
- \(y=\sinh ^{-1}x=\ln \big( x+ \sqrt{x^{2}+1}\big)\) for all real \(x\)
- \(y=\cosh ^{-1}x=\ln \big( x+ \sqrt{x^{2}-1}\big)\) \(x ≥ 1\)
- \(y=\tanh ^{-1}x=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)\) \(\vert x\vert < 1\)
- \(y=\coth ^{-1}x=\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right)\) \(\vert x\vert > 1\)
Basic Derivative Formulas (pp. 245, 248):
- \(\dfrac{d}{dx}\sinh x=\cosh x\)
- \(\dfrac{d}{dx}\text{sech}~x=-\text{sech}~x\tanh x\)
- \(\dfrac{d}{dx}\cosh =\sinh x\)
- \(\dfrac{d}{dx}\text{csch}~x=-\text{csch}~x\coth x\)
- \(\dfrac{d}{dx}\tanh x=\text{sech}^{2}x\)
- \(\dfrac{d}{dx}\coth x=-\text{csch}^{2}x\)
- \(\dfrac{d}{dx}\sinh ^{-1}x=\frac{1}{\sqrt{x^{2}+1}}\)
- \(\dfrac{d}{dx}\cosh ^{-1}x=\frac{1}{\sqrt{x^{2}-1}}\quad x > 1\)
- \(\frac{d}{dx}\tanh ^{-1}x=\frac{1}{1-x^{2}} \quad \vert x\vert < 1\)
Section |
You should be able to… |
Example |
Review Exercises |
3.1 |
1 Differentiate a composite function (p. 198) |
1-5 |
1, 13, 24 |
|
2 Differentiate \(y = a^{x},\) \(a > 0,\) \(a≠ 1\) (p. 202) |
6 |
19, 22 |
|
3 Use the Power Rule for functions to find a derivative (p. 202) |
7, 8 |
1, 11, 12, 14, 17 |
|
4 Use the Chain Rule for multiple composite functions (p. 204) |
9 |
15, 18, 61 |
3.2 |
1 Find a derivative using implicit differentiation (p. 209) |
1-4 |
43-52, 73, 81 |
|
2 Find higher-order derivatives using implicit differentiation (p. 212) |
5 |
49-52 |
|
3 Differentiate functions with rational exponents (p. 213) |
6, 7 |
2-8, 15, 16, 61-64 |
|
4 Find the derivative of an inverse function (p. 214) |
8 |
53, 54 |
|
5 Differentiate inverse trigonometric functions (p. 216) |
9, 10 |
32-38 |
3.3 |
1 Differentiate logarithmic functions (p. 222) |
1-3 |
20, 21, 23, 25-30, 52, 72 |
|
2 Use logarithmic differentiation (p. 225) |
4-7 |
9, 10, 31, 71 |
|
3 Express \(e\) as a limit (p. 227) |
8 |
55, 56 |
3.4 |
1 Find the differential of a function and interpret it geometrically (p. 230) |
1 |
65, 69, 70 |
|
2 Find the linear approximation to a function (p. 232) |
2 |
67 |
|
3 Use differentials in applications (p. 233) |
3, 4 |
66, 68 |
|
4 Use Newton’s Method to approximate a real zero of a function (p. 234) |
5, 6 |
78-80 |
3.5 |
1 Find a Taylor Polynomial (p. 240) |
1-3 |
74-77 |
3.6 |
1 Define the hyperbolic functions (p. 243) |
1 |
57, 58 |
|
2 Establish identities for hyperbolic functions (p. 244) |
2 |
59, 60 |
|
3 Differentiate hyperbolic functions (p. 245) |
3, 4 |
39-41 |
|
4 Differentiate inverse hyperbolic functions (p. 246) |
5-7 |
42 |