Question

How do you find the derivative of \[\dfrac{1}{{{t}^{2}}}\]?

Answer 1

Hint: This type of problem is based on the topic of derivation and the formula of derivatives. We will solve this problem using the simple formula of derivation. The formula is \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n\times {{x}^{n-1}}\] . In this type of question, first we will find the simplest form of \[\dfrac{1}{{{t}^{2}}}\]. Then, we will find the derivative of that equation.

Complete step by step answer:
Let us solve the problem.
It is given in the question that \[y=\dfrac{1}{{{t}^{2}}}\]
\[\dfrac{1}{{{t}^{2}}}\] can also be written in the form of \[{{t}^{-2}}\] .
Therefore, \[y=\dfrac{1}{{{t}^{2}}}\] can also be written as
\[y={{t}^{-2}}\]
Now, we will apply here the differentiation rule of the derivative \[y=\dfrac{1}{{{t}^{2}}}\] . Also, we can say that we have to find the derivative of \[y={{t}^{-2}}\] .
Applying the differentiation rule to the above derivative using the differentiation formula \[\dfrac{d}{dx}{{x}^{n}}=n\times {{x}^{n-1}}\] , we get
(After putting the value of x as t and n as -2 in the above formula)
\[\dfrac{d}{dx}\left( {{t}^{-2}} \right)=\left( -2 \right)\times {{t}^{(-2-1)}}\]
The above equation can also be written as
\[\dfrac{d}{dx}\left( {{t}^{-2}} \right)=\left( -2 \right)\times {{t}^{-3}}\]
\[\Rightarrow \dfrac{d}{dx}\left( {{t}^{-2}} \right)=-2{{t}^{-3}}\]
Hence, we get the derivative of \[{{t}^{-2}}\].
The derivative or the differentiation of \[{{t}^{-2}}\] is \[-2{{t}^{-3}}\] .

Note:
For solving this problem, we should have proper knowledge of derivation. We should know how to find the derivation of an equation. One should be aware of making mistakes in the derivation. Otherwise, the solution will always be wrong. One more method is there to solve the problem. For that method, another formula should be known. The formula is \[\dfrac{d}{dx}\left( \dfrac{u}{v} \right)=\dfrac{v\dfrac{d}{dx}u-u\dfrac{d}{dx}v}{{{v}^{2}}}\] . If we use this formula, then in this formula by putting the value of u as 1 and v as \[{{t}^{2}}\] . Then, we will the derivative and then the solution will be found.