5. Applications of Derivatives

Implicit Differentiation

5. Applications of Derivatives

Implicit Differentiation

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Multiple Choice

Find $dy/dx$ for the equation below using implicit differentiation.
$xy=\sin\left(y\right)$

$dy/dx=\frac{\sin\left(y\right)}{x}$

$dy/dx=\frac{x}{\cos\left(y\right)}$

$dy/dx=\frac{y}{\cos\left(y\right)-x}$

$dy/dx=\cos\left(y\right)$

Verified step by step guidance

Step 1: Start with the given equation xy = sin(y). This equation involves both x and y, so we will use implicit differentiation to find dy/dx.

Step 2: Differentiate both sides of the equation with respect to x. For the left-hand side, use the product rule: d(xy)/dx = x(dy/dx) + y. For the right-hand side, differentiate sin(y) with respect to x, which requires the chain rule: d(sin(y))/dx = cos(y) * (dy/dx).

Step 3: Combine the results from Step 2. The differentiated equation becomes: x(dy/dx) + y = cos(y) * (dy/dx).

Step 4: Isolate dy/dx on one side of the equation. Subtract y from both sides to get: x(dy/dx) = cos(y) * (dy/dx) - y.

Step 5: Factor out dy/dx from the terms on the right-hand side: dy/dx * (x - cos(y)) = -y. Finally, solve for dy/dx by dividing both sides by (x - cos(y)): dy/dx = -y / (x - cos(y)).