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 root sinx by chain rule

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.

Also Read:

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).

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