Chapter 3 - Derivatives - 3.4 The Product and Quotient Rules - 3.4 Exercises - Page 161: 72

$$\eqalign{ & {\bf{a}}.\,\,y = 20x - 32 \cr & {\bf{b}}.\,\,y = - 5x + 8 \cr} $$

$$\eqalign{ & {\bf{a}}. \cr & {\text{Let }}g\left( x \right) = {x^2}f\left( x \right) \cr & {\text{Calculate }}g\left( 2 \right) \cr & g\left( 2 \right) = {\left( 2 \right)^2}f\left( 2 \right) \cr & g\left( 2 \right) = {\left( 2 \right)^2}\left( 2 \right) \cr & g\left( 2 \right) = 8 \cr & {\text{Point }}\left( {2,8} \right) \cr & \cr & {\text{Differentiate}} \cr & g'\left( x \right) = \frac{d}{{dx}}\left[ {{x^2}f\left( x \right)} \right] \cr & g'\left( x \right) = {x^2}f'\left( x \right) + 2xf\left( x \right) \cr & {\text{Calculate }}g'\left( 2 \right) \cr & g'\left( 2 \right) = {\left( 2 \right)^2}f'\left( 2 \right) + 2\left( 2 \right)f\left( 2 \right) \cr & g'\left( 2 \right) = {\left( 2 \right)^2}\left( 3 \right) + 2\left( 2 \right)\left( 2 \right) \cr & g'\left( 2 \right) = 12 + 8 \cr & g'\left( 2 \right) = 20 \cr & {\text{Find an equation of the line tangent to }}x = 2 \cr & y - 8 = 20\left( {x - 2} \right) \cr & y - 8 = 20x - 40 \cr & y = 20x - 32 \cr & \cr & {\bf{b}}.\, \cr & {\text{Let }}h\left( x \right) = \frac{{f\left( x \right)}}{{x - 3}} \cr & {\text{Calculate }}h\left( 2 \right) \cr & h\left( 2 \right) = \frac{{f\left( 2 \right)}}{{2 - 3}} \cr & h\left( 2 \right) = \frac{2}{{2 - 3}} \cr & h\left( 2 \right) = - 2 \cr & {\text{Point }}\left( {2, - 2} \right) \cr & \cr & {\text{Differentiate}} \cr & h'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{f\left( x \right)}}{{x - 3}}} \right] \cr & h'\left( x \right) = \frac{{\left( {x - 3} \right)f'\left( x \right) - f\left( x \right)}}{{{{\left( {x - 3} \right)}^2}}} \cr & {\text{Calculate }}h'\left( 2 \right) \cr & h'\left( 2 \right) = \frac{{\left( {2 - 3} \right)f'\left( 2 \right) - f\left( 2 \right)}}{{{{\left( {2 - 3} \right)}^2}}} \cr & h'\left( 2 \right) = \frac{{\left( { - 1} \right)\left( 3 \right) - 2}}{{{{\left( { - 1} \right)}^2}}} \cr & h'\left( 2 \right) = - 5 \cr & {\text{Find an equation of the line tangent to }}x = 2 \cr & y + 2 = - 5\left( {x - 2} \right) \cr & y + 2 = - 5x + 10 \cr & y = - 5x + 8 \cr} $$