Question

Find the derivative of the following function:
${x^{ - 4}}\left( {3 - 4{x^{ - 5}}} \right)$

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
$y = {x^{ - 4}}\left( {3 - 4{x^{ - 5}}} \right)$
Now first simplify this so we have,
$y = 3{x^{ - 4}} - 4{x^{ - 4 - 5}}$
$y = 3{x^{ - 4}} - 4{x^{ - 9}}$
Now differentiate it w.r.t. x we have,
$ \Rightarrow \dfrac{d}{{dx}}y = \dfrac{d}{{dx}}\left[ {3{x^{ - 4}} - 4{x^{ - 9}}} \right]$
Now as we know that differentiation of $\dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = \left( { - n} \right){{\text{x}}^{ - n - 1}}$ so use this property in above equation we have,
$ \Rightarrow \dfrac{d}{{dx}}y = \left[ {3\left( { - 4} \right){x^{ - 4 - 1}} - 4\left( { - 9} \right){x^{ - 9 - 1}}} \right]$
Now simplify this equation we have,
$ \Rightarrow \dfrac{d}{{dx}}y = - 12{x^{ - 5}} + 36{x^{ - 10}}$
$ \Rightarrow \dfrac{d}{{dx}}y = 12{x^{ - 5}}\left[ { - 1 + 3{x^{ - 5}}} \right]$
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.