Question

Given that y=8/x⁴, find the value of dy/dx when x=2 . Hence estimate the value of 8/(1.99)⁴

Answer 1

Given:

`y=(8)/(x^4)`

Differentiate y with respect to x,

`\begin{gathered} (dy)/(dx)=(d)/(dx)((8)/(x^4)) \\ =(d)/(dx)(8x^(-4)) \\ \text{Apply rule of derivative,} \\ (d)/(dx)(x^n)=nx^(n-1) \\ (dy)/(dx)=8(-4)x^(-4-1) \\ (dy)/(dx)=-32x^(-5) \\ (dy)/(dx)=-(32)/(x^5) \end{gathered}`

When x = 2,

`\begin{gathered} (dy)/(dx)=-(32)/(2^5) \\ =-(32)/(32) \\ =-1 \end{gathered}`

Now, eastimate the value of,

`\begin{gathered} y=(8)/(x^4)\text{ at x=1.99} \\ y=(8)/((1.99)^4) \\ y=0.510\text{ } \end{gathered}`

Answer:

`\begin{gathered} (dy)/(dx)=-(32)/(x^5) \\ At\text{ x=2} \\ (dy)/(dx)=-1 \end{gathered}`