Question

Find the diviative of the following

`y = ( √(1 + 2x)) 5`
​

Answer 1

`(d)/(dx)\left(√(1\:+\:2x)\right)5 = \boxed{(5)/(√(1+2x))}`

Explanation:

Given
`y = \left(√(1\:+\:2x)\right)5`

we are asked to find
`(dy)/(dx)`

`(dy)/(dx) = (d)/(dx)\left(√(1\:+\:2x)\right)5\\\\= 5(d)/(dx)\left(√(1+2x)\right)\\\\`

Find
`(d)/(dx)\left(√(1+2x)\right)`:

`Let \;u = 1 + 2x\\\\f(u) = \sqrt(u)\\\\`

`\mathrm{Apply\:the\:chain\:rule}:\quad (df\left(u\right))/(dx)=(df)/(du)\cdot (du)/(dx)`

`= (d)/(du)\left(√(u)\right)(d)/(dx)\left(1+2x\right)`

`(d)/(du)\left(√(u)\right) = (d)/(du)\left(u^{(1)/(2)}\right)\\\\= (1)/(2)u^{(1)/(2)-1}\\\\= (1)/(2√(u))\\\\\\`

Substitute back u = 1 + 2x

`= (1)/(2√(1+2x))`

`(d)/(dx)(1 + 2x) =(d)/(dx)(1)} + (d)/(dx){2x}\\\\= 0 + 2 \\\\= 2\\`

Therefore

`(dy)/(dx) = (d)/(dx)\left(√(1\:+\:2x)\right)5\\\\= 5(d)/(dx)\left(√(1+2x)\right)\\\\`

`= 5\cdot (1)/(2√(1+2x))\cdot \:2\\\\= 5\cdot (1)/(√(1 + 2x))\\\\=(5)/(√(1+2x))`

Answer 2

Explanation:

1+2x)5

First you minus the 1 with the 5

Which you'll get a four then divide it by 2

Which you'll get x=2

But then times it by 5

Then you get y=10