Chapter 3 - Derivatives - 3.7 The Chain Rule - 3.7 Exercises - Page 193: 80

\[\begin{align} & \mathbf{a}\mathbf{.}\text{ }g\left( 1 \right)=4 \\ & \mathbf{b}\mathbf{.}\text{ }g'\left( x \right)=2x\cdot f'\left( {{x}^{2}} \right) \\ & \mathbf{c}\mathbf{.}\text{ }g'\left( 1 \right)=6 \\ & \mathbf{d}\mathbf{.}\text{ }y=6x-2 \\ \end{align}\]

\[\begin{align} & \text{Let }g\left( x \right)=f\left( {{x}^{2}} \right) \\ & \\ & \mathbf{a}\mathbf{.}\text{ Calculate }g\left( 1 \right) \\ & g\left( 1 \right)=f\left( {{\left( 1 \right)}^{2}} \right) \\ & g\left( 1 \right)=f\left( 1 \right) \\ & \text{The graph passes through the point }\left( 1,4 \right)\to f\left( 1 \right)=4,\text{ then} \\ & g\left( 1 \right)=4 \\ & \\ & \mathbf{b}\mathbf{.}\text{ Calculate }g'\left( x \right) \\ & g'\left( x \right)=\frac{d}{dx}\left[ f\left( {{x}^{2}} \right) \right] \\ & \text{By the chain rule} \\ & g'\left( x \right)=f'\left( {{x}^{2}} \right)\frac{d}{dx}\left[ {{x}^{2}} \right] \\ & g'\left( x \right)=f'\left( {{x}^{2}} \right)\left( 2x \right) \\ & g'\left( x \right)=2x\cdot f'\left( {{x}^{2}} \right) \\ & \\ & \mathbf{c}\mathbf{.}\text{ Calculate }g'\left( 1 \right) \\ & g'\left( x \right)=2x\cdot f'\left( {{x}^{2}} \right) \\ & g'\left( 1 \right)=2\left( 1 \right)\cdot f'\left( {{\left( 1 \right)}^{2}} \right) \\ & g'\left( 1 \right)=2f'\left( 1 \right) \\ & \text{The tangent line to the graph of }f\text{ at }\left( 1,4 \right)\text{ is }y=3x+1,\text{ recall} \\ & \text{that }\underbrace{y=3x+1}_{y=mx+b}\Rightarrow m=3,\text{ and the slope of the tangent line at} \\ & \text{the point }\left( x,y \right)\text{ is }m=f'\left( x \right),\text{ then }f'\left( 1 \right)=3,\text{ then} \\ & g'\left( 1 \right)=2f'\left( 1 \right) \\ & g'\left( 1 \right)=2\left( 3 \right) \\ & g'\left( 1 \right)=6 \\ & \\ & \mathbf{d}\mathbf{.}\text{ When }x=1\text{ the slope of the graph }g\left( x \right)\text{ is }m=g'\left( 1 \right) \\ & m=g'\left( 1 \right)=6 \\ & m=6 \\ & \text{Passes through the point }\left( 1,4 \right),\text{ then} \\ & y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\ & y-4=6\left( x-1 \right) \\ & y-4=6x-6 \\ & \text{ }y=6x-2 \\ \end{align}\]