Question

Please help!!! Identify the methods of differentiation, and then find the derivative using the methods. THANKS

Answer 1

First, notice that the function is the quotient between two expressions. Then, first we need to use the quotient rule to find the derivative:

`(d)/(dx)(g(x))/(h(x))=(h(x)\cdot(d)/(dx)g(x)-g(x)\cdot(d)/(dx)h(x))/(h(x)^2)`

In this case, g(x) = 3x+2 and h(x) = x^3. Then:

`\begin{gathered} (d)/(dx)f(x)=(d)/(dx)(3x+2)/(x^3) \\ =(x^3\cdot(d)/(dx)(3x+2)-(3x+2)\cdot(d)/(dx)x^3)/((x^3)^2) \end{gathered}`

Now, notice that the derivatives of 3x+2 and x^3 appear in the numerator. Use the power rule to find the derivative of those expressions:

`(d)/(dx)x^n=^{}nx^(n-1)`

Then:

`\begin{gathered} (d)/(dx)(3x+2)=3x^0+0=3\cdot1+0=3+0=3 \\ \\ (d)/(dx)x^3=3x^2 \end{gathered}`

So, the differentiation continues:

`\begin{gathered} (x^3\cdot(d)/(dx)(3x+2)-(3x+2)\cdot(d)/(dx)x^3)/((x^3)^2) \\ =(x^3\cdot(3)-(3x+2)\cdot(3x^2))/((x^3)^2) \end{gathered}`

Finally, simplify the expression:

`\begin{gathered} (x^3\cdot(3)-(3x+2)\cdot(3x^2))/((x^3)^2) \\ =(3x^3-(3x\cdot3x^2+2\cdot3x^2))/(x^6) \\ =(3x^3-(9x^3+6x^2))/(x^6) \\ =(3x^3-9x^3-6x^2)/(x^6) \\ =(-6x^3-6x^2)/(x^6) \\ =(-6x-6)/(x^4) \\ =(-6(x+1))/(x^4) \end{gathered}`

Therefore, we used both the power rule and the quotient rule to find the derivative, and the derivative is:

`(d)/(dx)(3x+2)/(x^3)=-(6(x+1))/(x^4)`