Suppose that we have a multivariate function $f(a,b,c,a_1,b_1)$, where $a,b,c,a_1,b_1$ are all real numbers.
$f = \frac{{c({a^2} + {b^2}) + {{({a^2} + {b^2})}^{1/2}}}}{{c{{({a^2} + {b^2})}^2}}}{a_1}{b_1}$
I generate a vector using the multivariate function $f(a,b,c,a_1,b_1)$. This is a set of evaluations of $f$ that I can plot as a 1D curve.
Taking the numerical partial derivative (i.e. Numerical differentiation), of the vector with respect to $a_1$ yields an approximation to:
$\frac{{\partial f}}{{\partial {a_1}}} = \frac{{c({a^2} + {b^2}) + {{({a^2} + {b^2})}^{1/2}}}}{{c{{({a^2} + {b^2})}^2}}}{b_1}$
However, suppose that all $a_1$ is unknown, but I do know $ca_1$, which is the product of $c$ with $a_1$. I also know $cb_1$, but not $a,b,c$ or $b_1$.
Under what conditions can I approximate the denominator of the numerical derivative using only $ca_1$? What are the error bounds associated with taking:
$\frac{{\partial f}}{{\partial (c{a_1})}}$
So rather than using the change in $a_1$, I would use some other type of change such as the change in $ca_1$.
I also know the ratio $a_1/b_1$ for all $a_1$ and $b_1$.
I also know $c{({a^2} + {b^2})^{1/2}}$ for all $c,a,b$
Can anything be done to approximate $a_1$, even using numerical or statistical methods? What additional information might be required?