Monday, September 26, 2022

$\sqrt{1+\sqrt{1+\sqrt{x}}}$ derivative | What is the derivative of $\frac{d}{dx}\sqrt{1+\sqrt{1+\sqrt{x}}}$?

Hello everyone. This lesson we will tell the derivative of $\sqrt{1+\sqrt{1+\sqrt{x}}}$.

What is the derivative of $\sqrt{1+\sqrt{1+\sqrt{x}}}$?

Compute the derivative of $\sqrt{1+\sqrt{1+\sqrt{x}}}$.

Here we have a more complicated chain of compositions, so we use the chain rule twice. At the outermost “layer” we have the function $g(x)=1+\sqrt{1+\sqrt{x}}$ plugged into $f(x)=\sqrt{x}$, so applying the chain rule once gives


Not we need the derivative of $\sqrt{1+\sqrt{x}}$. Using the chain rule again:


So the original derivative is



Using the chain rule, the power rule, and the product rule, it is possible to avıid using the quotient rule entirely.


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