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$F\left(x\right)\={\left(x^4\+3x^2\+1\right)}^2$$\stackrel{\text{assign as function}}{\to }$$F$

The derivative, simplified

$F\prime \left(x\right)$ = $2\left(x^4+3x^2+1\right)\left(4x^3+6x\right)$ = $4 x \left(2 x^2\+3\right) \left(x^4\+3 x^2\+1\right)$ 

Annotated stepwise solution

$\frac{\ⅆ}{\ⅆ x} F\left(x\right)$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Differentiation Steps}\\ & & \frac{\ⅆ}{\ⅆx}\left({\left(x^4+3x^2+1\right)}^2\right)\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{chain}}\text{rule to the term}{\left(x^4+3x^2+1\right)}^2\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{chain}}\text{rule}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)=\mathrm{f\text{'}}\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)\\ & \text{◦}& \text{Outside function}\\ & & f\left(v\right)=v^2\\ & \text{◦}& \text{Inside function}\\ & & g\left(x\right)=x^4+3x^2+1\\ & \text{◦}& \text{Derivative of outside function}\\ & & \frac{\ⅆ}{\ⅆv}f\left(v\right)=2v\\ & \text{◦}& \text{Apply composition}\\ & & \mathrm{f\text{'}}\left(g\left(x\right)\right)=2x^4+6x^2+2\\ & \text{◦}& \text{Derivative of inside function}\\ & & \frac{\ⅆ}{\ⅆx}g\left(x\right)=4x^3+6x\\ & \text{◦}& \text{Put it all together}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)=\left(2x^4+6x^2+2\right)\cdot \left(4x^3+6x\right)\\ & & \text{This gives:}\\ & & \left(2x^4+6x^2+2\right)\left(4x^3+6x\right)\end{array}$

Figure 2.4.1(a)   Context Panel access to Differentiation Rules

$F\prime \left(x\right)$

$\=\frac{d}{\mathrm{dx}}f\left(g\left(x\right)\right)$

$\=\frac{d}{\mathrm{dx}}f\left(z\left(x\right)\right)$

$\=\left(\frac{d}{\mathrm{dz}}\genfrac{}{}{0ex}{}{f\left(z\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{z\=g\left(x\right)}\right)\cdot \frac{\mathrm{dz}}{\mathrm{dx}}$

$\=\(\frac{d}{\mathrm{dz}}\genfrac{}{}{0ex}{}{f\left(z\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{z\=g\left(x\right)}\)\cdot g\prime \left(x\right)$

$F\prime \left(x\right)$

$\=f\prime \left(g\left(x\right)\right) g\prime \left(x\right)$

$\=2 g\left(x\right) g\prime \left(x\right)$

$\=2\left(x^4\+3 x^2\+1\right) \left(4 x^3\+6 x\right)$

$\=4 x \left(2 x^2\+3\right) \left(x^4\+3 x^2\+1\right)$