Questions: Calculate dy/dx using implicit differentiation cos y + 3 = x

Transcript text: Calculate $\frac{d y}{d x}$ using implicit differentiation \[ \cos y+3=x \]

Solution

Step 1: Differentiate both sides of the equation with respect to x

Given equation: $\cos{\left(y \right)} + 3 = x$

Differentiating with respect to $x$, we get:

$\frac{d}{dx}(\cos{\left(y \right)} + 3) = \frac{d}{dx}(x)$

This yields: $- \sin{\left(y \right)} \frac{d}{d x} y = 1$

Step 2: Solve for dy/dx

Solving for $\frac{dy}{dx}$, we find:

$\frac{dy}{dx} = - \frac{1}{\sin{\left(y \right)}}$

The derivative $\frac{dy}{dx}$ rounded to 5 decimal places is -1/sin(y).

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