Question

How do you find the derivative of `sqrt(3x+1)`?

Answer 1

First off, recall that the derivative of `sqrtu = 1/(2sqrtu)`. If you don't remember that, simply change the square root to a `+"1/2"` exponent and use the power rule: `d/(dx)[x^n] = nx^(n-1)`.

Now, in your expression, `u = 3x + 1`. Since `u` is a function of `x`, you must incorporate the chain rule:

`(df)/(dx) = (df)/(du)(du)/(dx)`

for `f(u) = sqrt(u(x)) = sqrt(3x + 1)`.

So, taking the derivative, we have:

`color(blue)(d/(dx)[sqrt(3x + 1)]) = 1/(2sqrt(u(x))) * (du)/(dx)`

`= 1/(2sqrt(3x + 1)) * d/(dx)[3x + 1]`

`= color(blue)(3/(2sqrt(3x + 1)))`