Question

If \[y = \dfrac{1}{{\left( {a - z} \right)}}\], then \[\dfrac{{dz}}{{dy}} = \]
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

Hint: In calculus, the term differentiation refers to the process of determining a function's derivative. We obtain the instantaneous rate of change of one variable with respect to another, which is why the process is termed differentiation.

Formula Used: \[\dfrac{{d\left( {\dfrac{1}{x}} \right)}}{{dx}} = - \dfrac{1}{{{x^2}}}\], that is, differentiation of \[\dfrac{1}{x}\] with respect to x is \[ - \dfrac{1}{{{x^2}}}\].

Complete step-by-step solution:
We have the given function as \[y = \dfrac{1}{{\left( {a - z} \right)}}\]. We will first rearrange the terms of the function, making z as the subject, and then we will find the derivative of that function with respect to y, which is \[\dfrac{{dz}}{{dy}}\].
We rearrange the terms of the function as,
\[
  y = \dfrac{1}{{\left( {a - z} \right)}}\\
   \Rightarrow a - z = \dfrac{1}{y}\\
   \Rightarrow a - \dfrac{1}{y} = z\\
   \Rightarrow z = a - \dfrac{1}{y} ......(1)
 \]

We will apply the distribution law of differentiation to find the derivative of the function with respect to y in equation (1) as,
\[
  z = a - \dfrac{1}{y} \\
   \Rightarrow \dfrac{{dz}}{{dy}} = \dfrac{{d\left( {a - \dfrac{1}{y}} \right)}}{{dy}} \\
   \Rightarrow \dfrac{{dz}}{{dy}} = \dfrac{{da}}{{dy}} - \dfrac{{d\left( {\dfrac{1}{y}} \right)}}{{dy}} \\
   \Rightarrow \dfrac{{dz}}{{dy}} = 0 - \left( { - \dfrac{1}{{{y^2}}}} \right)
 \]

Further Simplifying, we get,
\[\dfrac{{dz}}{{dy}} = \dfrac{1}{{{y^2}}} ......(2)\]

Substitute the value \[y = \dfrac{1}{{\left( {a - z} \right)}}\] in equation (2) as,
\[
  \dfrac{{dz}}{{dy}} = \dfrac{1}{{{y^2}}} \\
   \Rightarrow \dfrac{{dz}}{{dy}} = \dfrac{1}{{{{\left( {\dfrac{1}{{\left( {a - z} \right)}}} \right)}^2}}} \\
   \Rightarrow \dfrac{{dz}}{{dy}} = {\left( {\dfrac{1}{{\left( {\dfrac{1}{{\left( {a - z} \right)}}} \right)}}} \right)^2} \\
   \Rightarrow \dfrac{{dz}}{{dy}} = {\left( {a - z} \right)^2}
 \]

So, option A \[{\left( {a - z} \right)^2}\] is the required solution.

Additional Information: We can always apply the distribution law while differentiating a function if there is more than one term that is separated by a plus or a minus sign. If the terms are separated by a multiplication or division sign then we will have to apply the chain rule.

Note: When we differentiate any function, always check the terms in the function whether they are the variables of the given function or some other constant or any other variable and not to include them while differentiating. When we replace a value with a variable make sure that the other constants remain the same as they were before.