Question

Find the derivative of the following function:
${x^5}\left( {3 - 6{x^{ - 9}}} \right)$

Answer 1

Hint: In this question first simplify the function later on use the property of differentiation which is given as $\dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = \left( { - n} \right){{\text{x}}^{ - n - 1}}$ so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Let
$y = {x^5}\left( {3 - 6{x^{ - 9}}} \right)$
Now first simplify this so we have,
$y = 3{x^5} - 6{x^{5 - 9}}$
$y = 3{x^5} - 6{x^{ - 4}}$
Now differentiate it w.r.t. x we have,
$ \Rightarrow \dfrac{d}{{dx}}y = \dfrac{d}{{dx}}\left[ {3{x^5} - 6{x^{ - 4}}} \right]$
Now as we know that differentiation of $\dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = \left( { - n} \right){{\text{x}}^{ - n - 1}}{\text{ and }}\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$ so use this property in above equation we have,
$ \Rightarrow \dfrac{d}{{dx}}y = \left[ {3 \times 5{x^{5 - 1}} - 6\left( { - 4} \right){x^{ - 4 - 1}}} \right]$
Now simplify this equation we have,
$ \Rightarrow \dfrac{d}{{dx}}y = 15{x^4} + 24{x^{ - 5}}$
So this is the required differentiation.

Note – Whenever we face such types of questions the key concept is always recall the formula of ${x^n}$ differentiation which is stated above then first simplify the given function then use the property of differentiation as above and again simplify we will get the required answer.