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Differentiate sinsqrt(x) + cos^(2)sqrt(x...

Differentiate `sinsqrt(x) + cos^(2)sqrt(x)`

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To differentiate the function \( y = \sin(\sqrt{x}) + \cos^2(\sqrt{x}) \), we will follow these steps: ### Step 1: Differentiate the function We start with the function: \[ y = \sin(\sqrt{x}) + \cos^2(\sqrt{x}) \] Now, we differentiate both sides with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx}(\sin(\sqrt{x})) + \frac{d}{dx}(\cos^2(\sqrt{x})) \] ### Step 2: Differentiate \( \sin(\sqrt{x}) \) Using the chain rule, we differentiate \( \sin(\sqrt{x}) \): \[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \] Now, we know that: \[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \] So, \[ \frac{d}{dx}(\sin(\sqrt{x})) = \cos(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} = \frac{\cos(\sqrt{x})}{2\sqrt{x}} \] ### Step 3: Differentiate \( \cos^2(\sqrt{x}) \) Using the chain rule and the power rule, we differentiate \( \cos^2(\sqrt{x}) \): \[ \frac{d}{dx}(\cos^2(\sqrt{x})) = 2\cos(\sqrt{x}) \cdot \frac{d}{dx}(\cos(\sqrt{x})) \] Now, we differentiate \( \cos(\sqrt{x}) \) using the chain rule: \[ \frac{d}{dx}(\cos(\sqrt{x})) = -\sin(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) = -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \] Thus, \[ \frac{d}{dx}(\cos^2(\sqrt{x})) = 2\cos(\sqrt{x}) \cdot \left(-\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) = -\frac{\sin(\sqrt{x}) \cos(\sqrt{x})}{\sqrt{x}} \] ### Step 4: Combine the derivatives Now we combine the derivatives we found: \[ \frac{dy}{dx} = \frac{\cos(\sqrt{x})}{2\sqrt{x}} - \frac{\sin(\sqrt{x}) \cos(\sqrt{x})}{\sqrt{x}} \] We can factor out \( \frac{1}{\sqrt{x}} \): \[ \frac{dy}{dx} = \frac{1}{\sqrt{x}} \left( \frac{\cos(\sqrt{x})}{2} - \sin(\sqrt{x}) \cos(\sqrt{x}) \right) \] ### Step 5: Simplify the expression We can express \( -\sin(\sqrt{x}) \cos(\sqrt{x}) \) as \( -\frac{1}{2} \sin(2\sqrt{x}) \) using the double angle identity: \[ \frac{dy}{dx} = \frac{1}{\sqrt{x}} \left( \frac{\cos(\sqrt{x})}{2} - \frac{1}{2} \sin(2\sqrt{x}) \right) \] ### Final Answer Thus, the derivative of the function is: \[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \left( \cos(\sqrt{x}) - \sin(2\sqrt{x}) \right) \]

To differentiate the function \( y = \sin(\sqrt{x}) + \cos^2(\sqrt{x}) \), we will follow these steps: ### Step 1: Differentiate the function We start with the function: \[ y = \sin(\sqrt{x}) + \cos^2(\sqrt{x}) \] Now, we differentiate both sides with respect to \( x \): ...
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