Chapter Review

• Definition Derivative of function at a number (form 1) \(f^\prime (c) =\lim\limits_{x\rightarrow c} \dfrac{f( x) -f( c) }{x-c}\) provided the limit exists. (p. 150)

Three Interpretations of the Derivative

• Physical interpretation When the distance \(s\) at time \(t\) of an object in rectilinear motion is given by the function \(s=f( t) \), the derivative \(f^\prime ( t_{0}) \) is the velocity of the object at time \(t_{0}\). (pp. 145-146)
• Geometric interpretation If \(y=f( x)\), the derivative \(f^\prime ( c) \) is the slope of the tangent line to the graph of \(f\) at the point \(\left( c,f( c) \right)\). (pp. 147-148)
• Rate of change of a function interpretation If \(y=f( x)\), the derivative \(f^\prime ( c)\) is the rate of change of \(f\) at \(c\). (pp. 149-150)

• Definition of a derivative function (form 3) (p. 154) \(f^\prime ( x) =\lim\limits_{h\rightarrow 0}\dfrac{f( x+h) -f( x) }{h},\) provided the limit exists.
• Theorem If a function \(f\) has a derivative at a number \(c\), then \(f\) is continuous at \(c\). (p. 158)
• Corollary If a function \(f\) is not continuous at a number \(c\), then \(f\) has no derivative at \(c\). (p. 159)

• Leibniz notation \(\dfrac{dy}{dx}=\dfrac{d}{dx}y=\dfrac{d}{dx}f( x) \) (p. 163)

• Basic derivatives

• \(\dfrac{d}{dx}A=0\) A is a constant (p. 163)
• \(\dfrac{d}{dx}x=1\) (p. 164)
• \(\dfrac{d}{dx}e^{x}=e^{x}\) (p. 169)

Properties of Derivatives

• Simple Power Rule \(\dfrac{d}{dx}x^{n}=nx^{n-1},\) \(n\) an integer (pp. 164 and 177)
• Sum Rule \[ \begin{eqnarray*} \dfrac{d}{dx}[ f+g] &=& \dfrac{d}{dx}f+\dfrac{d }{dx}g\\ ( f+g)^\prime &=&f^\prime +g^\prime (\text{p}.166) \end{eqnarray*} \]
• Difference Rule \[ \begin{eqnarray*} \dfrac{d}{dx}[ f-g] &=&\dfrac{d}{dx} f- \dfrac{d}{dx}g\\ ( f-g)^\prime &=&f^\prime -g^\prime (\text{p}. 167) \end{eqnarray*} \]
• Constant Multiple Rule \[ \begin{eqnarray*} \dfrac{d}{dx}[ kf] &=&k\dfrac{d}{dx }f\\ ( kf)^\prime &=&k\cdot f^\prime \end{eqnarray*} \] \(k\) is a constant (p. 165)

Properties of Derivatives

• Product Rule \[ \begin{eqnarray*} \dfrac{d}{dx}( fg) &=&f\left( \dfrac{d}{dx} g\right) +\left( \dfrac{d}{dx}f\right) g\\ ( f g)^\prime &=&fg^\prime +f^\prime g \end{eqnarray*} \] (p. 174)
• Quotient Rule \[ \begin{eqnarray*} \dfrac{d}{dx}\left( \dfrac{f}{g}\right) &=&\dfrac{ \left( \dfrac{d}{dx}f\right) g-f\!\left( \dfrac{d}{dx}g\right) }{g^{2}}\\ \left( \dfrac{f}{g}\right)^{\!\!\prime }&=&\dfrac{f^\prime g-fg^\prime }{ g^{2}}, \end{eqnarray*} \] provided \(g( x) \neq 0\) (p. 175)
• Reciprocal Rule \[ \begin{eqnarray*} \dfrac{d}{dx}\left( \dfrac{1}{g}\right) &=& -\dfrac{\dfrac{d}{dx}g}{g^{2}}\\ \left( \dfrac{1}{g}\right)^{\!\!\prime } &=&-\dfrac{g^\prime }{g^{2}}, \end{eqnarray*} \] provided \(g( x) \neq 0\) (p. 176)
• Higher-order derivatives

  Prime notation: \(f'' ( x)\), \(f''' ( x)\), \(f^{( 4) }( x) ,\ldots,\) \(f^{( n) }( x) \)

  Liebniz notation: \(\dfrac{d^{2}}{dx^{2}}[ f( x) ] ,\) \(\dfrac{ d^{3}}{dx^{3}}\left[ f( x) \right] ,\) \(\dfrac{d^{4}}{dx^{4}}[ f( x) ] , \ldots , \dfrac{d^{n}}{dx^{n}}[ f( x) ] \) (p. 178)

Basic Derivatives

• \(\dfrac{d}{dx}\sin x=\cos x\) (p. 186)
• \(\dfrac{d}{dx}\cos x=-\sin x\) (p. 187)
• \(\dfrac{d}{dx}\tan x=\sec ^{2}x\) (p. 188)
• \(\dfrac{d}{dx}\sec x=\sec x\tan x\) (p. 188)
• \(\dfrac{d}{dx}\csc x=-\csc x\cot x\) (p. 188)
• \(\dfrac{d}{dx}\cot x=-\csc ^{2}x\) (p. 188)