Question

Finding a Derivative, find the derivative of the algebraic function.

Answer 1

We are given the function:

`f(x)=x^4(1-(2)/(x+1))`

To find its derivative, we can distribute x^4 first so that we won't have to apply both the product rule and the quotient rule.

`f(x)=x^4-(2x^4)/(x+1)`

So we'll just solve for the derivative of the first term, x^4, then add it to the derivative of the second term, 2x^4/(x+1).

`f^(\prime)(x)=g^(\prime)(x)+h^(\prime)(x)`

where g(x) = x^4 and h(x) = 2x^4/(x+1).

`\begin{gathered} g(x)=x^4 \\ g^(\prime)(x)=4x^(4-1) \\ g^(\prime)(x)=4x^3 \end{gathered}`
`\begin{gathered} h(x)=(2x^4)/(x+1) \\ \\ h^(\prime)(x)=((x+1)(2)(4)(x^(4-1))-(2x^4)(1+0))/((x+1)^2) \\ \\ h^(\prime)(x)=((x+1)(8x^3)-(2x^4))/((x+1)^2) \\ \\ h^(\prime)(x)=(8x^4+8x^3-2x^4)/((x+1)^2) \\ \\ h^(\prime)(x)=(6x^4+8x^3)/((x+1)^2) \\ \\ h^(\prime)(x)=(2x^3(3x+4))/((x+1)^2) \end{gathered}`

So f'(x) must be:

`f^(\prime)(x)=4x^3+(2x^(3)(3x+4))/((x+1)^(2))`