Chapter Review

Printed Page 250

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)

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)

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)

• 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)

  251

• 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):

• $\dfrac{d}{dx}\sinh x=\cosh x$
• $\dfrac{d}{dx}\text{sech}~x=-\text{sech}~x\tanh x$
• $\dfrac{d}{dx}\cosh =\sinh x$
• $\dfrac{d}{dx}\text{csch}~x=-\text{csch}~x\coth x$
• $\dfrac{d}{dx}\tanh x=\text{sech}^{2}x$
• $\dfrac{d}{dx}\coth x=-\text{csch}^{2}x$
• $\dfrac{d}{dx}\sinh ^{-1}x=\frac{1}{\sqrt{x^{2}+1}}$
• $\dfrac{d}{dx}\cosh ^{-1}x=\frac{1}{\sqrt{x^{2}-1}}\quad x > 1$
• $\frac{d}{dx}\tanh ^{-1}x=\frac{1}{1-x^{2}} \quad \vert x\vert < 1$