# Derivative of root sinx by Chain Rule

The derivative of root sinx is equal to cosx/(2√sinx).  In this post, we will learn how to differentiate square root of sinx by the chain rule of derivatives.

## Derivative of Square Root of Sin x by Chain Rule

Question: Find the Derivative of  $\sqrt{\sin x}$ by the chain rule.

Solution:

To find the derivative of root sinx by the chain rule, we will follow the steps provided below.

Step 1: Let z=sinx

Differentiating both sides with respect to x, we get that

$\dfrac{dz}{dx} = \cos x$

Step 2: By the chain rule of derivatives,

$\dfrac{d}{dx}(\sqrt {\sin x})$ = $\dfrac{d}{dx}(\sqrt z)$

= $\dfrac{d}{dz}(\sqrt z) \times \dfrac{dz}{dx}$

= $\dfrac{d}{dz}(z^{1/2}) \times \cos x$

= ½ z½ -1 × cosx by the power rule of derivatives d/dx(xn)=nxn-1.

Step 3: Putting z=sinx and simplifying we get that

$\dfrac{d}{dx}(\sqrt {\sin x})$ = $\dfrac{\cos x}{2 \sqrt{z}}$ as z-1/2=1/√z.

= $\dfrac{\cos x}{2 \sqrt{\sin x}}$

So the derivative of root sinx is equal to $\dfrac{\cos x}{2 \sqrt{\sin x}}$ and this is obtained by the chain rule of derivatives.

Derivative of $\sqrt{\sin x}$ by First Principle

Derivative of $\sqrt{\cos x}$

Derivative of $\sqrt{e^x}$

## FAQs

Q1: Find the Derivative root sinx.

Answer: cosx/(2 √sin x) is the derivative of the square root of sinx, that is, d/dx(√sinx) = cosx/(2√sin x).