Find the slope of tangent to Y = 4 / x + 2 √(x) at x=4 ​

Question

Find the slope of tangent to
Y =

`4 / x + 2 √(x)`
at x=4
​

Answer 1

slope =
`(1)/(4)`

Explanation:

To find the slope, differentiate with respect to x and evaluate at x = 4

`(dy)/(dx)` is the measure of the slope of the tangent at x = a

Differentiate each term using the power rule

`(d)/(dx)` (a
`x^(n)` ) = na
`x^(n-1)`

Given

y =
`(4)/(x)` + 2
`√(x)`

= 4
`x^(-1)` + 2
`x^{(1)/(2) }`, hence

`(dy)/(dx)` = - 4
`x^(-2)` +
`x^{-(1)/(2) }`

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

At x = 4

`(dy)/(dx)` = -
`(4)/(16)` +
`(1)/(√(4) )`

= -
`(1)/(4)` +
`(1)/(2)` =
`(1)/(4)`