Math / Calculus

QuestionUse implicit differentiation to find $\frac{d y}{d x}$ . $\begin{array}{l} (3 x y+4)^{2}=6 y \\ \frac{d y}{d x}=\square \end{array}$ $\square$

Studdy Solution

STEP 1

1. We are given an equation involving both $x$ and $y$ .
2. We need to use implicit differentiation to find $\frac{dy}{dx}$ .
3. Both $x$ and $y$ are functions of each other.

STEP 2

1. Differentiate both sides of the equation with respect to $x$ .
2. Solve for $\frac{dy}{dx}$ .

STEP 3

Differentiate both sides of the equation with respect to $x$ . Use the chain rule and implicit differentiation for the left side:
Given: $(3xy + 4)^2 = 6y$
Differentiate the left side: $\frac{d}{dx}((3xy + 4)^2) = 2(3xy + 4) \cdot \frac{d}{dx}(3xy + 4)$
Apply the product rule to $3xy$ : $\frac{d}{dx}(3xy) = 3 \left( x \frac{dy}{dx} + y \right)$
So, the derivative of the left side becomes: $2(3xy + 4)(3x \frac{dy}{dx} + 3y)$
Differentiate the right side: $\frac{d}{dx}(6y) = 6 \frac{dy}{dx}$

STEP 4

Set the derivatives equal to each other:
$2(3xy + 4)(3x \frac{dy}{dx} + 3y) = 6 \frac{dy}{dx}$

STEP 5

Solve for $\frac{dy}{dx}$ . First, expand and simplify the left side:
$6(3xy + 4)(x \frac{dy}{dx} + y) = 6 \frac{dy}{dx}$
Distribute the 6: $6(3xy + 4)x \frac{dy}{dx} + 6(3xy + 4)y = 6 \frac{dy}{dx}$
Rearrange terms to isolate $\frac{dy}{dx}$ : $18x(3xy + 4) \frac{dy}{dx} - 6 \frac{dy}{dx} = -18y(3xy + 4)$
Factor out $\frac{dy}{dx}$ : $\frac{dy}{dx} (18x(3xy + 4) - 6) = -18y(3xy + 4)$
Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{-18y(3xy + 4)}{18x(3xy + 4) - 6}$
Simplify the expression if possible: $\frac{dy}{dx} = \frac{-18y(3xy + 4)}{18x(3xy + 4) - 6}$
The derivative $\frac{dy}{dx}$ is:
$\boxed{\frac{-18y(3xy + 4)}{18x(3xy + 4) - 6}}$

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