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$F\left(x\right)\={\left(x^4\+1\right)}^5$$\stackrel{\text{assign as function}}{\to }$$F$

The derivative

$F″\left(x\right)$ = $320{\left(x^4\+1\right)}^3{x}^6\+60{\left(x^4\+1\right)}^4{x}^2$$\stackrel{\text{factor}}{\=}$$20x^2\left(19x^4\+3\right){\left(x^4\+1\right)}^3$$$

Annotated stepwise calculation of the second derivative

$\frac{\ⅆ}{\ⅆ x} F\prime \left(x\right)$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Differentiation Steps}\\ & & \frac{\ⅆ}{\ⅆx}\left(20{\left(x^4+1\right)}^4{x}^3\right)\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\frac{\ⅆ}{\ⅆx}\left(20{\left(x^4+1\right)}^4{x}^3\right)\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \frac{\ⅆ}{\ⅆx}\left(Cf\left(x\right)\right)=C\frac{\ⅆ}{\ⅆx}f\left(x\right)\\ & \text{◦}& \text{This means:}\\ & & \frac{\ⅆ}{\ⅆx}\left(20{\left(x^4+1\right)}^4{x}^3\right)=20\cdot \frac{\ⅆ}{\ⅆx}\left({\left(x^4+1\right)}^4{x}^3\right)\\ & & \text{We can rewrite the derivative as:}\\ & & 20\frac{\ⅆ}{\ⅆx}\left({\left(x^4+1\right)}^4{x}^3\right)\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{product}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{product}}\text{rule}\\ & & \frac{\ⅆ}{\ⅆx}\left(f\left(x\right)g\left(x\right)\right)=\frac{\ⅆ}{\ⅆx}f\left(x\right)g\left(x\right)+f\left(x\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)\\ & & f\left(x\right)={\left(x^4+1\right)}^4\\ & & g\left(x\right)=x^3\\ & & \text{This gives:}\\ & & 20\frac{\ⅆ}{\ⅆx}\left({\left(x^4+1\right)}^4\right)x^3+20{\left(x^4+1\right)}^4\frac{\ⅆ}{\ⅆx}\left(x^3\right)\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{power}}\text{rule to the term}\frac{\ⅆ}{\ⅆx}\left(x^3\right)\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \frac{\ⅆ}{\ⅆx}\left(x^n\right)=n{x}^{n-1}\\ & \text{◦}& \text{This means:}\\ & & \frac{\ⅆ}{\ⅆx}\left(x^3\right)=3\cdot x^2\\ & & \text{We can rewrite the derivative as:}\\ & & 20\frac{\ⅆ}{\ⅆx}\left({\left(x^4+1\right)}^4\right)x^3+60{\left(x^4+1\right)}^4{x}^2\\ \text{▫}& & \text{4. Apply the}\mathbf{\text{chain}}\text{rule to the term}{\left(x^4+1\right)}^4\\ & \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^4\\ & \text{◦}& \text{Inside function}\\ & & g\left(x\right)=x^4+1\\ & \text{◦}& \text{Derivative of outside function}\\ & & \frac{\ⅆ}{\ⅆv}f\left(v\right)=4v^3\\ & \text{◦}& \text{Apply composition}\\ & & \mathrm{f\text{'}}\left(g\left(x\right)\right)=4{\left(x^4+1\right)}^3\\ & \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)=4{\left(x^4+1\right)}^3\cdot \left(4x^3\right)\\ & & \text{This gives:}\\ & & 320{\left(x^4+1\right)}^3{x}^6+60{\left(x^4+1\right)}^4{x}^2\end{array}$

Figure 2.4.4(a)   Context Panel access to Differentiation Rules

$F″\left(x\right)$

$\=\frac{d}{\mathrm{dx}}\left(F\prime \left(x\right)\right)$

$\=\frac{d}{\mathrm{dx}}\left(20 x^3 {\left(x^4\+1\right)}^4\right)$

$\=20 \frac{d}{\mathrm{dx}}\left(x^3 {\left(x^4\+1\right)}^4\right)$

$\=20\left\{x^3\left[4\cdot {\left(x^4\+1\right)}^3\cdot 4 x^3\right]\+{\left(x^4\+1\right)}^4\cdot 3 x^2\right\}$

$\=20x^2\left(19x^4\+3\right){\left(x^4\+1\right)}^3$