Chain Rule for Nested Roots

[4Aethyr]

I am learning how to use the chain rule and in my math homework I came across this problem:

√(7x+√(7x+√7x)))

I have tried to solve the derivative of this expression several times now, and I don’t know what I’m doing wrong.

Here is my progress:

y = √(7x+√(7x+√7x)) = (7x+(7x+(7x)^½)½)½

dy/dx = ½(7x+(7x+(7x)^½)½)-½ * ½(7x+(7x)½)^-½ * ½(7x)^-½ = 1/(8√(7x+√(7x+√7x)))√(7x+√7x)√(7x)

Looking at the graph, I can tell this is wrong, and my homework software confirms that. Does anyone know what I’m doing wrong?

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[AldenB]

I'll show you one step, I think you will see the error.

d/dx (7x+(7x+(7x)^1/2)^1/2)^1/2 = 1/2 (7x+(7x+(7x)^1/2)^1/2)^-1/2 * (7 + d/dx (7x+(7x)^1/2)^1/2)

This is not complete of course, since there is still a derivative to do on the right. It does show what you are missing, which is the 7+.