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Question: If\(y = \dfrac{{1 - x}}{{{x^2}}}\), then \(\dfrac{{dy}}{{dx}}\)is A. \(\dfrac{2}{{{x^2}}} + \dfrac...

Ify=1−xx2y = \dfrac{{1 - x}}{{{x^2}}}, then dydx\dfrac{{dy}}{{dx}}is
A. 2x2+2x3\dfrac{2}{{{x^2}}} + \dfrac{2}{{{x^3}}}
B. −2x3+1x2 - \dfrac{2}{{{x^3}}} + \dfrac{1}{{{x^2}}}
C. −2x2+2x2 - \dfrac{2}{{{x^2}}} + \dfrac{2}{{{x^2}}}
D. None of these

Explanation

Solution

First, we shall analyze the given information so that we are able to solve the problem. Generally in Mathematics, the derivative refers to the rate of change of a function with respect to a variable. Here in this question, we are asked to calculate the first derivative of the given equation. First, we need to split the given equation for our convenience. Then, we need to differentiate the resultant equation. Here, we are applying the power rule of differentiation to find the required answer.
Formula to be used:
The formula for the power rule of differentiation is as follows.
ddx(xn)=nxn−1\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}

Complete step by step answer:
It is given that y=1−xx2y = \dfrac{{1 - x}}{{{x^2}}}
We are asked to calculate the derivative of yy
To find:dydx\dfrac{{dy}}{{dx}}
y=1−xx2y = \dfrac{{1 - x}}{{{x^2}}}
=1x2−xx2= \dfrac{1}{{{x^2}}} - \dfrac{x}{{{x^2}}} (Here we splitted the terms)
⇒y=1x2−1x\Rightarrow y = \dfrac{1}{{{x^2}}} - \dfrac{1}{x}
Now, we shall differentiate the above equation with respect to x.
Thus,dydx=ddx(1x2−1x)\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{1}{{{x^2}}} - \dfrac{1}{x}} \right)
We need to separate the terms on the right side of the above equation.
The formula for the power rule of differentiation is as follows.
ddx(xn)=nxn−1\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}
That is dydx=ddx(1x2)−ddx(1x)\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{1}{{{x^2}}}} \right) - \dfrac{d}{{dx}}\left( {\dfrac{1}{x}} \right)
=ddx(x−2)−ddx(x−1)= \dfrac{d}{{dx}}\left( {{x^{ - 2}}} \right) - \dfrac{d}{{dx}}\left( {{x^{ - 1}}} \right)
=−2x−2−1−(−1)x−1−1= - 2{x^{ - 2 - 1}} - \left( { - 1} \right){x^{ - 1 - 1}} (Here we applied the power rule of differentiation)
=−2x−3+x−2= - 2{x^{ - 3}} + {x^{ - 2}}
=−2x3+1x2= - \dfrac{2}{{{x^3}}} + \dfrac{1}{{{x^2}}}
Hence dydx=−2x3+1x2\dfrac{{dy}}{{dx}} = \dfrac{{ - 2}}{{{x^3}}} + \dfrac{1}{{{x^2}}} .

So, the correct answer is “Option B”.

Note: We often use the power rule to calculate the derivative of a variable raised to a power and the power rule is the most commonly used derivative rule. When we are asked to find the derivation of the given equation, we need to change the given equation for our convenience. Then we need to analyze where we need to apply the derivative formulae and where we need to apply the rule of differentiation while differentiating the given equation.