Question

If $y=\frac{1}{(a-z)}$, then $\frac{dz}{dy}=$

• A. ${\left(a-z\right)}^2$
• B. $-{\left(z-a\right)}^2$
• C. ${\left(z+a\right)}^2$
• D. $-{\left(z+a\right)}^2$

Answer 1

The correct option is A

${\left(a-z\right)}^2$

Explanation for the correct option :

Step 1: Express $z$ in terms of $y$.

$\begin{array}{rcl}y& =& \frac{1}{(a-z)}\\ & \Rightarrow & ay-yz=1\\ \Rightarrow z& =& a-\frac{1}{y}........\left(1\right)\end{array}$

Step 2: Find $\frac{dz}{dy}$

By differentiating $\left(1\right)$ w.r.t $y$, we get

$\begin{array}{rcl}\frac{dz}{dy}& =& \frac{d}{dy}\left(a-\frac{1}{y}\right)\\ & =& 0+\frac{1}{y^2}\\ & =& \frac{1}{\left({\displaystyle \frac{1}{{\left(a-z\right)}^2}}\right)}\\ & =& {\left(a-z\right)}^2\end{array}$

Hence, option A is correct.