exercise - differentiation but a test on your reasoning ♡

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

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

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

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

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

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

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

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

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

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

wiki ♡