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$F\left(x\right)\={\left(x^4\+1\right)}^{20}$$\stackrel{\text{assign as function}}{\to }$$F$

The derivative

$F\prime \left(x\right)$ = $80{\left(x^4+1\right)}^{19}{x}^3$

Annotated stepwise derivative

$\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+1\right)}^{20}\right)\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{chain}}\text{rule to the term}{\left(x^4+1\right)}^{20}\\ & \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^{20}\\ & \text{◦}& \text{Inside function}\\ & & g\left(x\right)=x^4+1\\ & \text{◦}& \text{Derivative of outside function}\\ & & \frac{\ⅆ}{\ⅆv}f\left(v\right)=20v^{19}\\ & \text{◦}& \text{Apply composition}\\ & & \mathrm{f\text{'}}\left(g\left(x\right)\right)=20{\left(x^4+1\right)}^{19}\\ & \text{◦}& \text{Derivative of inside function}\\ & & \frac{\ⅆ}{\ⅆx}g\left(x\right)=4x^3\\ & \text{◦}& \text{Put it all together}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)=20{\left(x^4+1\right)}^{19}\cdot \left(4x^3\right)\\ & & \text{This gives:}\\ & & 80{\left(x^4+1\right)}^{19}{x}^3\end{array}$

Figure 2.4.2(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)$

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

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

$\=20 g{\left(x\right)}^{19} g\prime \left(x\right)$

$\=20{\left(x^4\+1\right)}^{19} \left(4 x^3\right)$

$\=80 x^3 {\left(x^4\+1\right)}^{19}$