Question

Find dy over dxA) 2x+1 over x power 4 + 2

Answer 1

Explanation:

Given y defined below:

`y=(2x+1)/(x^4+2)`

To find the derivative of y, we use the quotient rule.

`(d)/(dx)((u)/(v))=(v(du)/(dx)-u(dv)/(dx))/(v^2)`

In the function:

`\begin{gathered} u=2x+1\implies(du)/(dx)=2 \\ v=x^4+2\implies(dv)/(dx)=4x^3 \end{gathered}`

Substitute these values into the formula:

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

We then simplify:

`\begin{gathered} (dy)/(dx)=(2x^4+4-8x^4-4x^3)/((x^4+2)^2) \\ =(2x^4-8x^4-4+4x^3)/((x^4+2)^2) \\ =(-6x^4+4x^3-4)/((x^4+2)^2) \\ =(-2(3x^4+2x^3-2))/((x^4+2)^2) \end{gathered}`

Answer:

The derivative dy/dx of the given function is:

`(dy)/(dx)=(-2(3x^(4)+2x^(3)-2))/((x^(4)+2)^(2))`