Question

Given the equation of a curve is y=5/x²The value of dy/dx= -10/27Hence, estimate the value of 5/(2.98)²

Answer 1

The given equation is:

`y=(5)/(x^2)`

a. Find the value of dy/dx when x=3

Start by finding the derivative:

`(dy)/(dx)=(d((5)/(x^2)))/(dx)`

You also can express 5/x^2 as 5*x^(-2):

`(5)/(x^2)=5x^(-2)`

You know the derivative of a power is:

`(d)/(dx)x^n=n\cdot x^(n-1)`

Apply it to your case:

`(d)/(dx)5x^(-2)=(-2)\cdot5\cdot x^(-2-1)=-10\cdot x^(-3)`

And finally:

`\begin{gathered} x^(-n)=(1)/(x^n) \\ \text{Apply it to your equation} \\ (dy)/(dx)=(-10)/(x^3) \end{gathered}`
`\begin{gathered} \text{When x=3} \\ (dy)/(dx)=(-10)/(3^3)=(-10)/(27) \end{gathered}`

b. Estimate the value of 5/(2.98)^2:

`\begin{gathered} y=(5)/(x^2)=(5)/(2.98^2)=0.563 \\ \text{Which also means x=2.98} \\ \text{Let's find the derivative when x=2.98} \\ (dy)/(dx)=(-10)/(2.98^3)=(-10)/(26.46)=-0.377 \end{gathered}`