In the above example we have introduced the variables \( a \) and \( b \), and our function is
$$ f(x) = f(a(x)) = b= \exp{a}, $$with \( a=x^2 \). We can decompose the derivative of \( f \) with respect to \( x \) as
$$ \frac{df}{dx}=\frac{df}{db}\frac{db}{da}\frac{da}{dx}. $$We note that since \( b=f(x) \) that
$$ \frac{df}{db}=1, $$leading to
$$ \frac{df}{dx}=\frac{db}{da}\frac{da}{dx}=2x\exp{x^2}, $$as before.