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

Find the derivative of the following function:
x5(3−6x−9){x^5}\left( {3 - 6{x^{ - 9}}} \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=x5(3−6x−9)y = {x^5}\left( {3 - 6{x^{ - 9}}} \right)
Now first simplify this so we have,
y=3x5−6x5−9y = 3{x^5} - 6{x^{5 - 9}}
y=3x5−6x−4y = 3{x^5} - 6{x^{ - 4}}
Now differentiate it w.r.t. x we have,
⇒ddxy=ddx[3x5−6x−4]\Rightarrow \dfrac{d}{{dx}}y = \dfrac{d}{{dx}}\left[ {3{x^5} - 6{x^{ - 4}}} \right]
Now as we know that differentiation of ddx(x−n)=(−n)x−n−1 and ddx(xn)=nxn−1\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,
⇒ddxy=[3×5x5−1−6(−4)x−4−1]\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,
⇒ddxy=15x4+24x−5\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 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.