Math  /  Calculus

QuestionUse implicit differentiation to find dydx\frac{\mathrm{dy}}{\mathrm{dx}}. (3xy+4)2=12ydydx=\begin{array}{l} (3 x y+4)^{2}=12 y \\ \frac{d y}{d x}=\square \end{array} \square

Studdy Solution

STEP 1

1. The given equation is (3xy+4)2=12y(3xy + 4)^2 = 12y.
2. We need to use implicit differentiation to find dydx\frac{dy}{dx}.

STEP 2

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

STEP 3

Differentiate the left side of the equation (3xy+4)2(3xy + 4)^2 with respect to xx using the chain rule. Let u=3xy+4u = 3xy + 4, then differentiate u2u^2 with respect to uu and multiply by the derivative of uu with respect to xx:
ddx[(3xy+4)2]=2(3xy+4)ddx(3xy+4)\frac{d}{dx}[(3xy + 4)^2] = 2(3xy + 4) \cdot \frac{d}{dx}(3xy + 4)
Now differentiate 3xy+43xy + 4 using the product rule:
ddx(3xy+4)=3(xdydx+y)\frac{d}{dx}(3xy + 4) = 3 \left( x \frac{dy}{dx} + y \right)
Thus, the derivative of the left side is:
2(3xy+4)3(xdydx+y)2(3xy + 4) \cdot 3 \left( x \frac{dy}{dx} + y \right)

STEP 4

Differentiate the right side of the equation 12y12y with respect to xx:
ddx[12y]=12dydx\frac{d}{dx}[12y] = 12 \frac{dy}{dx}

STEP 5

Set the derivatives from Step 1 and Step 2 equal to each other:
2(3xy+4)3(xdydx+y)=12dydx2(3xy + 4) \cdot 3 \left( x \frac{dy}{dx} + y \right) = 12 \frac{dy}{dx}

STEP 6

Simplify and solve for dydx\frac{dy}{dx}. First, expand the left side:
6(3xy+4)(xdydx+y)=12dydx6(3xy + 4) \left( x \frac{dy}{dx} + y \right) = 12 \frac{dy}{dx}
Distribute:
18x(3xy+4)dydx+6(3xy+4)y=12dydx18x(3xy + 4) \frac{dy}{dx} + 6(3xy + 4)y = 12 \frac{dy}{dx}
Rearrange terms to isolate dydx\frac{dy}{dx}:
18x(3xy+4)dydx12dydx=6(3xy+4)y18x(3xy + 4) \frac{dy}{dx} - 12 \frac{dy}{dx} = -6(3xy + 4)y
Factor out dydx\frac{dy}{dx}:
(18x(3xy+4)12)dydx=6(3xy+4)y\left(18x(3xy + 4) - 12\right) \frac{dy}{dx} = -6(3xy + 4)y
Solve for dydx\frac{dy}{dx}:
dydx=6(3xy+4)y18x(3xy+4)12\frac{dy}{dx} = \frac{-6(3xy + 4)y}{18x(3xy + 4) - 12}
The derivative dydx\frac{dy}{dx} is:
dydx=6(3xy+4)y18x(3xy+4)12\frac{dy}{dx} = \frac{-6(3xy + 4)y}{18x(3xy + 4) - 12}

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