exercise - differentiation but a test on your reasoning

worked example: Differentiate $f(x)={\left(1+(1+x^2{)}^3\right)}^4$

this is a hard question to test your skills - we have to use the chain rule many times

when solving something hard, we write down things step by step

$m(x)=1+x^2$
$n(x)=x^3$
$j(x)=(1+x{)}^4$
$k(x)=n(m(x))$

$f(x)=j(k(x))$

$f^{\prime }(x)=j^{\prime }(k(x))\cdot k^{\prime }(x)$

$j^{\prime }(x)=(4)(1+x{)}^3\cdot (1)$
$j^{\prime }(x)=4(x+1{)}^3$

$k^{\prime }(x)=n^{\prime }(m(x))\cdot m^{\prime }(x)$
$k^{\prime }(x)=(3)(1+x^2{)}^2\cdot 2x$
$k^{\prime }(x)=6x(1+x^2{)}^2$

$f^{\prime }(x)=j^{\prime }((1+x^2{)}^3)\cdot 6x(1+x^2{)}^2$

$f^{\prime }(x)=4((1+x^2{)}^3+1{)}^3\cdot 6x(1+x^2{)}^2$

$f^{\prime }(x)=24x(1+x^2{)}^2((1+x^2{)}^3+1{)}^3$