What is chain rule for second order derivative of a composite function?

Question

I have a function for example like this: $f(g(h(i(j))))$.

I know first derivative can be found as $f'(j)=df/dg * dg/dh * dh/di * di/dj$.

What would be second order derivative of $f$ wrt $j$ be; $f''(j)$? based on chain rule?

Help really needed and I would be really thankful for any.

Answer 1

It involves the product rule and the chain rule.

For example, $$ y=f(g(h(x)))$$

$$ y'= f'(g(h(x))\cdot g'(h(x))\cdot h'(x)$$ For the second derivative we have to use product rule as well as the chain rule.

$$ y'' = f''(g(h(x))\cdot g'(h(x))\cdot h'(x)\cdot g'(h(x))\cdot h'(x)\\+f'(g(h(x))\cdot g''(h(x))\cdot h'(x)\cdot h'(x) \\+f'(g(h(x))\cdot g'(h(x))\cdot h''(x)$$ We may simplify the answer, to get

$$y'' = f''(g(h(x))\cdot g'^2(h(x))\cdot h'^2(x)\\ +f'(g(h(x))\cdot g''(h(x))\cdot h'^2(x) \\+f'(g(h(x))\cdot g'(h(x))\cdot h''(x)$$