implicit xy+4y=10 — step-by-step solution

⚠ xy is read as the product x*y because y appears elsewhere

For

x y + 4 y = 10

, treat y as a differentiable function of x and differentiate both sides

Set F(x, y) =

x y + 4 y - 10

. The chain rule gives Fx + Fy*dy/dx = 0

For Fx, differentiate with respect to x while holding y constant

⇥Let's perform the differentiation:

\frac{d}{d x} (x y + 4 y - 10)

⇥

\frac{d}{d x} (x y) + \frac{d}{d x} (4 y) + 0

⇥ℹ

\frac{d}{d x} (x y)

y

‖

\frac{d}{d x} (4 y)

0

⇥

y + 0

⇥

y

For Fy, differentiate with respect to y while holding x constant

⇥Let's perform the differentiation:

\frac{d}{d y} (x y + 4 y - 10)

⇥

\frac{d}{d y} (x y) + \frac{d}{d y} (4 y) + 0

⇥ℹ

\frac{d}{d y} (x y)

x

‖

\frac{d}{d y} (4 y)

4

⇥

x + 4

The partial derivatives are Fx =

y

and Fy =

x + 4

x + 4 \neq 0

, isolate the derivative: dy/dx = -Fx/Fy. This condition is taken on the original curve

▶Implicit derivative dy/dx:

- \frac{y}{x + 4}