Derivative of $x \over x+k$?

Question

This seems like it should be easy but its got me beat:

$$c = {a \over {a + b} }$$

What is:

$$\partial c \over \partial a$$

?

Answer 1

• Quotient rule on $\cfrac x{x+k}$
• Product rule on $x(x+k)^{-1}$
• Chain rule on $\left (1+\cfrac kx\right)^{-1}$

Answer 2

$a/(a+b)= 1- b/(a+b)$ Thus the derivative is $b(a+b)^-2$