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.
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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
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).