Question

How do I find the derivative of $y=\dfrac{1}{{{x}^{8}}}$?

Answer 1

This type of problem is based on the concept of differentiation. First, we have to consider the whole function and make some necessary calculations, that is, multiplying ${{x}^{-8}}$ in both the numerator and denominator, to bring x term alone in the numerator and I in the denominator. Then, use power rule of differentiation, that is, $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$ in the obtained function. Here, n=-8. Therefore, we need to differentiate ${{x}^{-8}}$ separately and thus, the differentiation of the given function is obtained.

Complete step-by-step answer:
According to the question, we are asked to find the derivative of the given function.
We have been given the function $y=\dfrac{1}{{{x}^{8}}}$. -----(1)
We first have to find the differentiation of $\dfrac{1}{{{x}^{8}}}$ .
We know that, on multiplying the same term on the numerator and denominator of a function, the value of the function does not change.
Therefore, on multiplying ${{x}^{-8}}$ on both the numerator and denominator of function (1), we get
$y=\dfrac{1}{{{x}^{8}}}\times \dfrac{{{x}^{-8}}}{{{x}^{-8}}}$
On further simplification, we get,
$y=\dfrac{{{x}^{-8}}}{{{x}^{8}}\times {{x}^{-8}}}$.
We know that, ${{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}$.
Using this power rule in the above obtained expression, we get,
$y=\dfrac{{{x}^{-8}}}{{{x}^{8-8}}}$
$\Rightarrow y=\dfrac{{{x}^{-8}}}{{{x}^{0}}}$ --------(2)
We know that, ${{x}^{0}}=1$.
Using this property in equation (2), we get,
$y=\dfrac{{{x}^{-8}}}{1}$
On further simplification, we get,
$y={{x}^{-8}}$ --------(3)
Now, we have to differentiate the obtained function $y={{x}^{-8}}$.
We have to use the power rule of differentiation, that is, $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$.
On comparing with expression (3), we get n=-8.
Therefore,
$\dfrac{d}{dx}\left( {{x}^{-8}} \right)=\left( -8 \right){{x}^{-8-1}}$
On further simplification, we get,
$\dfrac{d}{dx}\left( {{x}^{-8}} \right)=\left( -8 \right){{x}^{-9}}$
$\therefore \dfrac{d}{dx}\left( {{x}^{-8}} \right)=-8{{x}^{-9}}$
Hence, the derivative of $y=\dfrac{1}{{{x}^{8}}}$ is $-8{{x}^{-9}}$.

Note: Whenever you get this type of problem, we should always try to make the necessary changes in the given function to get the final solution of the function which will be the required answer. We should avoid calculation mistakes based on sign conventions. we can also this problem by using division rule of differentiation, that is, $\dfrac{d}{dx}\left( \dfrac{u}{v} \right)=\dfrac{v\dfrac{d}{dx}\left( u \right)-u\dfrac{d}{dx}\left( v \right)}{{{v}^{2}}}$. Here, u=1 and $v={{x}^{8}}$. Differentiate v using product rule considering n=8 and differentiation of 1 is 0. Substitute these values in the formula and obtain the required solution.