Question

A curve has equation y=x√x.find the equation of the tangent to the curve at the point(1, 5)

Answer 1

y - 5 =
`(3)/(2)`(x - 1)

Explanation:

Note that
`(dy)/(dx)` =
`m_(tangent)`

Differentiate using the power rule

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

Given

y = x
`√(x)` = x.
`x^{(1)/(2) }` =
`x^{(3)/(2) }`, then

`(dy)/(dx)` =
`(3)/(2)`
`x^{(1)/(2) }`

When x = 1

`(dy)/(dx)` =
`(3)/(2)` . 1 =
`(3)/(2)`

The equation of a line in point- slope form is

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

Here m =
`(3)/(2)` and (a, b) = (1, 5), thus

y - 5 =
`(3)/(2)`(x - 1) ← equation of tangent