use implicit differentiation to find y, and then evaluate y for x²y - 4x² = 0 at the point (2,4). y = □…

use implicit differentiation to find y, and then evaluate y for x²y - 4x² = 0 at the point (2,4). y = □ y|(2,4) = □ (simplify your answer.)

use implicit differentiation to find y, and then evaluate y for x²y - 4x² = 0 at the point (2,4). y = □ y|(2,4) = □ (simplify your answer.)

Answer

Explanation:

Step1: Differentiate both sides

Differentiate $x^{2}y - 4x^{2}=0$ with respect to $x$. Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ for $x^{2}y$ where $u = x^{2}$ and $v = y$. $\frac{d}{dx}(x^{2}y)-\frac{d}{dx}(4x^{2})=\frac{d}{dx}(0)$ $(2xy + x^{2}y^\prime)-8x = 0$

Step2: Solve for $y^\prime$

Isolate $y^\prime$ terms. $x^{2}y^\prime=8x - 2xy$ $y^\prime=\frac{8x - 2xy}{x^{2}}=\frac{2x(4 - y)}{x^{2}}=\frac{2(4 - y)}{x}$

Step3: Evaluate $y^\prime$ at $(2,4)$

Substitute $x = 2$ and $y = 4$ into $y^\prime$. $y^\prime|_{(2,4)}=\frac{2(4 - 4)}{2}=0$

Answer:

$y^\prime=\frac{2(4 - y)}{x}$ $y^\prime|_{(2,4)}=0$