Use implicit differentiation to find $\frac{d y}{d x}$ .
$e^{3 x y} = 7 y$
$\frac{d y}{d x} = ◻$

Answer

Expert–verified

Hide Steps

Answer

$\frac{d y}{d x} = \frac{- 3 y \cdot e^{3 x y}}{3 x \cdot e^{3 x y} - 7}$

Steps

Step 1 ：Differentiate both sides of the equation $e^{3 x y} = 7 y$ with respect to $x$

Step 2 ：Apply the chain rule to the left side: $\frac{d}{d x} (e^{3 x y}) = 3 y \cdot e^{3 x y} \cdot \frac{d x}{d x} + 3 x \cdot e^{3 x y} \cdot \frac{d y}{d x}$

Step 3 ：Differentiate the right side: $\frac{d}{d x} (7 y) = 7 \cdot \frac{d y}{d x}$

Step 4 ：Combine the derivatives: $3 y \cdot e^{3 x y} + 3 x \cdot e^{3 x y} \cdot \frac{d y}{d x} = 7 \cdot \frac{d y}{d x}$

Step 5 ：Isolate terms with $\frac{d y}{d x}$ : $3 x \cdot e^{3 x y} \cdot \frac{d y}{d x} - 7 \cdot \frac{d y}{d x} = - 3 y \cdot e^{3 x y}$

Step 6 ：Factor out $\frac{d y}{d x}$ : $\frac{d y}{d x} (3 x \cdot e^{3 x y} - 7) = - 3 y \cdot e^{3 x y}$

Step 7 ：Solve for $\frac{d y}{d x}$ : $\frac{d y}{d x} = \frac{- 3 y \cdot e^{3 x y}}{3 x \cdot e^{3 x y} - 7}$

Step 8 ： $\frac{d y}{d x} = \frac{- 3 y \cdot e^{3 x y}}{3 x \cdot e^{3 x y} - 7}$

Use implicit differentiation to find dydx.e3xy=7ydydx=◻

Answer

Popular Problems

Use implicit differentiation to find $\frac{d y}{d x}$ .
$e^{3 x y} = 7 y$
$\frac{d y}{d x} = ◻$