How do I find the derivative of this equation?

Question

d/dx 1+x / √x

Answer 1

If you divide first, then you don't need the quotient rule or the product rule.

(1+x) / √x = 1/√x + x/√x = x^-1/2 + x^1/2

So, the derivative is (-1/2)x^-3/2 + (1/2)x^-1/2

Answer 2

Hello Kelly!

To find the derivative of f(x) = (1+x) / √x we can using either the derivative Product Rule or the Quotient Rule. I'll use the product rule here and walk you through my process.

Recall that the product rule allows us to analyze the derivative of a function which is the product of two terms ("u" and "v") as shown below:

f(x) = u * v

d(f(x))/dx = u*[dv/dx] + v*[du/dx]

How can we apply product rule to our problem here? What would our u and v terms be?

We can visualize it more easily if we write our function like so...

f(x) = [1+x] * [1/√x]

u = 1 + x

v = 1 / √x

Now we need to find the derivatives of u and v seperately... Note that v = 1 / √x can also be equivalently expressed as v = x^-1/2, which may be less intimidating to differentiate!

u = 1 + x

du/dx = 1

v = x^-1/2

dv/dx = (-1/2)*x^-3/2

Now let's plug these values into our product rule formula...

d(f(x))/dx = u*[dv/dx] + v*[du/dx]

d(f(x))/dx = (1 + x)*[(-1/2)*x^-3/2] + (x^-1/2)*[1] (distribute terms...)

d(f(x))/dx = -(1/2)*x^-3/2 - (1/2)x^-1/2 + x^-1/2 (combine same order terms...)

d(f(x))/dx = -(1/2)*x^-3/2 + (1/2)x^-1/2

If desired we can do further algebraic simplification to combine the two terms into a single fractional term and get...

d(f(x))/dx = (x - 1) / (2x^3/2)

If you are more familiar with a "quotient rule" you could also apply it to this problem, but anytime that you *could* do quotient rule you can always go the product rule route instead if you prefer!

Hope that helps! Let me know if you have any questions,

--Zach