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

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

Explanation

Solution

Hint: In this question apply the product rule of differentiation which is given as ddx(uv)=uddxv+vddxu\dfrac{d}{{dx}}\left( {uv} \right) = u\dfrac{d}{{dx}}v + v\dfrac{d}{{dx}}u later on apply the formula of differentiation of ddx(x−n)=(−n)x−n−1 and ddx(constant)=0 and ddx(nx)=n\dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = \left( { - n} \right){{\text{x}}^{ - n - 1}}{\text{ and }}\dfrac{d}{{dx}}\left( {{\text{constant}}} \right) = 0{\text{ and }}\dfrac{d}{{dx}}\left( {nx} \right) = n so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Let
y=x−3(5+3x)y = {x^{ - 3}}\left( {5 + 3x} \right)
Now differentiate it w.r.t. x we have,
⇒ddxy=ddx[x−3(5+3x)]\Rightarrow \dfrac{d}{{dx}}y = \dfrac{d}{{dx}}\left[ {{x^{ - 3}}\left( {5 + 3x} \right)} \right]
Now here we use product rule of differentiate which is given as
ddx(uv)=uddxv+vddxu\dfrac{d}{{dx}}\left( {uv} \right) = u\dfrac{d}{{dx}}v + v\dfrac{d}{{dx}}u so use this property in above equation we have,
⇒ddxy=x−3ddx(5+3x)+(5+3x)ddx(x−3)\Rightarrow \dfrac{d}{{dx}}y = {x^{ - 3}}\dfrac{d}{{dx}}\left( {5 + 3x} \right) + \left( {5 + 3x} \right)\dfrac{d}{{dx}}\left( {{x^{ - 3}}} \right)
Now as we know that differentiation of ddx(x−n)=(−n)x−n−1 and ddx(constant)=0 and ddx(nx)=n\dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = \left( { - n} \right){{\text{x}}^{ - n - 1}}{\text{ and }}\dfrac{d}{{dx}}\left( {{\text{constant}}} \right) = 0{\text{ and }}\dfrac{d}{{dx}}\left( {nx} \right) = n so use this property in above equation we have,
⇒ddxy=x−3(0+3)+(5+3x)(−3x−3−1)\Rightarrow \dfrac{d}{{dx}}y = {x^{ - 3}}\left( {0 + 3} \right) + \left( {5 + 3x} \right)\left( { - 3{x^{ - 3 - 1}}} \right)
Now simplify this equation we have,
⇒ddxy=3x−3−15x−4−9x−3\Rightarrow \dfrac{d}{{dx}}y = 3{x^{ - 3}} - 15{x^{ - 4}} - 9{x^{ - 3}}
⇒ddxy=−15x−4−6x−3=−3x−3(5x−1+2)\Rightarrow \dfrac{d}{{dx}}y = - 15{x^{ - 4}} - 6{x^{ - 3}} = - 3{x^{ - 3}}\left( {5{x^{ - 1}} + 2} \right)
So this is the required differentiation.

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