Find Derivative

Question

Find derivative of y=sqrt(x+sqrt(x))

Give clear steps and explanations

Answer 1

Hi Jake,

Hopefully I'm interpreting your problem correctly. I think it's:

y=√(x+√x))

If that's the case, we're gonna have to use the chain rule. Let's start by letting u = x + √x.

∴ y = √u, and dy = d/du (√u) · d/dx (u)

For the derivative of √u, we use the power rule, where we multiply the expression by the power and then subtract one from it.

d/du (√u) = d/du (u^1/2) = (1/2)u^-1/2 = (1/2)(x+√x)^-1/2

Now, d/dx (u) = d/dx (x + √x) = d/dx (x) + d/dx (√x)

d/dx(x) = 1 and d/dx(√x) = (1/2)x^-1/2.

Putting this all together, we get the following:

dy/dx = d/du(√u) * du/dx = (1/2)(x+√x)^-1/2⋅(1+(1/2)x^-1/2), and that is your final answer. Not much simplification can be done outside of maybe putting the first bit on the bottom of the fraction (since its exponent is negative). Please let me know if I can clarify anything: Typing this is much less convenient than writing so I understand it may be a bit confusing.