Find dy/dx for the given functions: 1) y = √(4 + 2x) 2) y = (4 + sin²(3x))⁵ 3) x² + cot(xy) = 17 4) y = cot⁻¹(e^(5x))

Steps to Solve

1. Find the derivative of ( y = \frac{1}{\sqrt{4 + 2x}} )

Use the chain rule. Start by rewriting the function as:

$$ y = (4 + 2x)^{-1/2} $$

Then apply the chain rule:

$$ y' = -\frac{1}{2} (4 + 2x)^{-3/2} \cdot (2) $$

2. Differentiate ( y = (4 + \sin^2(3x))^5 )

Use the chain rule again. Let ( u = 4 + \sin^2(3x) ):

$$ y = u^5 $$

Then:

$$ y' = 5u^4 \cdot \frac{du}{dx} $$

Next, find ( \frac{du}{dx} ):

$$ \frac{du}{dx} = 2\sin(3x)(3\cos(3x)) = 6\sin(3x)\cos(3x) $$

Combine results:

$$ y' = 5(4 + \sin^2(3x))^4 \cdot 6\sin(3x)\cos(3x) $$

3. Differentiate ( x^2 + \cot(xy) = 17 )

Apply implicit differentiation. Differentiate each term:

$$ 2x + \frac{d}{dx}[\cot(xy)] = 0 $$

Using the product rule on ( \cot(xy) ), we get:

$$ -\csc^2(xy)(y + x y') = 0 $$

Solve for ( y' ):

$$ 2x - \csc^2(xy)(y + xy') = 0 $$

4. Differentiate ( y = \cot^{-1}(e^{5x}) )

Use the chain rule:

$$ y' = -\frac{1}{1 + (e^{5x})^2} \cdot \frac{d}{dx}(e^{5x}) $$

Calculate:

$$ \frac{d}{dx}(e^{5x}) = 5e^{5x} $$

So:

$$ y' = -\frac{5e^{5x}}{1 + e^{10x}} $$