Question

Find the gradient of the curve y = x^5/2 at the point where x = 4


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Answer 1

Given:

The equation of the curve is:

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

To find:

The gradient of the given curve at the point where
`x = 4`.

Solution:

We have,

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

Differentiate with respect to x.

`y'=(5)/(2)x^{(5)/(2)-1}`
`[\because (d)/(dx)x^n=nx^(n-1)]`

`y'=(5)/(2)x^{(3)/(2)}`

Substituting
`x=4`, we get

`y'=(5)/(2)(4)^{(3)/(2)}`

`y'=(5)/(2)(2^2)^{(3)/(2)}`

Using properties of exponents, we get

`y'=(5)/(2)(2)^{2* (3)/(2)}`
`[\because (a^m)^n=a^(mn)]`

`y'=(5)/(2)(2)^(3)`

`y'=(5)/(2)(8)`

`y'=20`

Therefore, the gradient of the given curve at the point where
`x = 4` is 20.