The derivative of cos root x is equal to (-sin√x)/2√x. Here we will find the derivative of cos√x by the chain rule of derivatives.

The differentiation of cos(√x) with respect to x is given by

$\dfrac{d}{dx}(\cos \sqrt{x})=-\dfrac{\sin \sqrt{x}}{2\sqrt{x}}$.

Let us now find the derivative of cos root x using the chain rule of differentiation.

## Derivative of cos(√x) by Chain Rule

To find the derivative of cos root x, we will use the chain rule of differentiation. Let us assume that

z = √x.

Differentiating, dz/dx = d/dx(x^{1/2}) = 1/2 x^{1-1/2} = 1/(2√x), by the power rule.

Now,

$\dfrac{d}{dx}(\cos \sqrt{x})$

= $\dfrac{d}{dx}(\cos z)$

= $\dfrac{d}{dz}(\cos z) \times \dfrac{dz}{dx}$, by the chain rule.

= $-\sin z \times \dfrac{1}{2\sqrt{x}}$ as we have dz/dx = 1/(2√x).

= $-\dfrac{\sin \sqrt{x}}{2\sqrt{x}}$ as z=√x.

So the derivative of cos root x is equal to (-sin√x)/2√x, and this is obtained by the chain rule of differentiation.

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## FAQs

### Q1: What is the derivative of cos root x?

Answer: The derivative of cos root x is (-sin√x)/2√x.

### Q2: If y= cos √x, then find dy/dx.

Answer: If y=cos√x, then dy/dx = (-sin√x)/2√x.