Question

How do you find the derivative of $\dfrac{1}{{{x}^{5}}}$?

Answer 1

In this question we have to find the derivative of a fraction which has the term ${{x}^{5}}$ in the denominator therefore, we will first take the reciprocal of the term using the property of exponents which is $\dfrac{1}{{{a}^{n}}}={{a}^{-n}}$ so that it comes in the numerator then we will use the formula of derivative which is ${{x}^{n}}=n{{x}^{n-1}}$ and simplify the expression to get the final solution.

Complete step by step answer:
We have the given expression as $\dfrac{1}{{{x}^{5}}}$
Since we have to find the derivative of the term, we can write is as:
$\Rightarrow \dfrac{d}{dx}\dfrac{1}{{{x}^{5}}}$
Now on using the property of exponents $\dfrac{1}{{{a}^{n}}}={{a}^{-n}}$, we can write the expression as:
$\Rightarrow \dfrac{d}{dx}{{x}^{-5}}$
Now we know that $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ so on using the formula on the expression, we get:
$\Rightarrow -5{{x}^{-5-1}}$
On simplifying the exponents, we get:
$\Rightarrow -5{{x}^{-6}}$
Now on rearranging the expression, we get:
$\Rightarrow -\dfrac{5}{{{x}^{6}}}$, which is the required solution.

Note: For these types of questions the formulas for the derivatives of the terms should be remembered.
It is to be remembered that the inverse of the number, also called the reciprocal of the number is the number dividing $1$, for example the reciprocal of $a$ is $\dfrac{1}{a}$, and it can also be expressed in terms of power as ${{a}^{-1}}$.
The inverse of the derivative is the integration and vice versa. If the derivative of a term $a$ is $b$, then the integration of the term $b$ will be $a$.
The chain rule is not required in this solution because we do not have a composite function. A composite function is a function in the form of $f(g(x))$ and it is solved using the formula $F'(x)=f'(g(x))g'(x)$ .