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

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

Explanation

Solution

Hint: In this question first simplify the function later on use the property of differentiation which is given as ddx(x−n)=(−n)x−n−1\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(3−4x−5)y = {x^{ - 4}}\left( {3 - 4{x^{ - 5}}} \right)
Now first simplify this so we have,
y=3x−4−4x−4−5y = 3{x^{ - 4}} - 4{x^{ - 4 - 5}}
y=3x−4−4x−9y = 3{x^{ - 4}} - 4{x^{ - 9}}
Now differentiate it w.r.t. x we have,
⇒ddxy=ddx[3x−4−4x−9]\Rightarrow \dfrac{d}{{dx}}y = \dfrac{d}{{dx}}\left[ {3{x^{ - 4}} - 4{x^{ - 9}}} \right]
Now as we know that differentiation of ddx(x−n)=(−n)x−n−1\dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = \left( { - n} \right){{\text{x}}^{ - n - 1}} so use this property in above equation we have,
⇒ddxy=[3(−4)x−4−1−4(−9)x−9−1]\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,
⇒ddxy=−12x−5+36x−10\Rightarrow \dfrac{d}{{dx}}y = - 12{x^{ - 5}} + 36{x^{ - 10}}
⇒ddxy=12x−5[−1+3x−5]\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 xn{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.