Chapter 2 - The Derivative - 2.2 The Derivative Function - Exercises Set 2.2 - Page 131: 10

$$\,\,\,f'\left( x \right) = - \frac{2}{{{x^3}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y = 2x + 3$$

$$\eqalign{ & f\left( x \right) = \frac{1}{{{x^2}}};\,\,\,\,a = - 1 \cr & {\text{Evaluate at }}a = - 1 \cr & f\left( { - 1} \right) = \frac{1}{{{{\left( { - 1} \right)}^2}}} \cr & f\left( { - 1} \right) = 1 \cr & f\left( { - 1} \right) = 1 \Rightarrow \left( { - 1,1} \right) \cr & {\text{The function }}f'\left( x \right){\text{ is defined by the formula}} \cr & \,\,\,f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h} \cr & {\text{Then}} \cr & \,\,\,f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{{{\left( {x + h} \right)}^2}}} - \frac{1}{{{x^2}}}}}{h} \cr & \,\,\,f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{{x^2} + 2xh + {h^2}}} - \frac{1}{{{x^2}}}}}{h} \cr & \,\,\,f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{{{x^2} - {x^2} - 2xh - {h^2}}}{{{x^2}\left( {{x^2} + 2xh + {h^2}} \right)}}}}{h} \cr & \,\,\,f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{ - 2xh - {h^2}}}{{h{x^2}\left( {{x^2} + 2xh + {h^2}} \right)}} \cr & \,\,\,f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{ - 2x - h}}{{{x^2}\left( {{x^2} + 2xh + {h^2}} \right)}} \cr & h \to 0 \cr & \,\,\,f'\left( x \right) = \frac{{ - 2x - 0}}{{{x^2}\left( {{x^2} + 2x\left( 0 \right) + {{\left( 0 \right)}^2}} \right)}} \cr & \,\,\,f'\left( x \right) = \frac{{ - 2x}}{{{x^2}\left( {{x^2}} \right)}} \cr & \,\,\,f'\left( x \right) = - \frac{2}{{{x^3}}} \cr & \cr & {\text{Calculate the tangent line }}y = f\left( x \right){\text{ at }}x = a \cr & \,\,\,m = f'\left( { - 1} \right) = - \frac{2}{{{{\left( { - 1} \right)}^3}}} \cr & \,\,\,m = 2 \cr & {\text{The tangent line is given by }}y - {y_1} = m\left( {x - {x_1}} \right){\text{ at }}\left( { - 1,1} \right) \cr & \,\,y - 1 = 2\left( {x + 1} \right) \cr & \,\,y - 1 = 2x + 2 \cr & \,\,\,\,\,\,\,\,\,\,y = 2x + 3 \cr & \cr & \,\,\,f'\left( x \right) = - \frac{2}{{{x^3}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y = 2x + 3 \cr} $$