Differentiate x2 y + y3 = 1, y + x2 = 3, and e^xy = x + y using implicit differentiation.

Question

ScanSolve AI Tutor · Accepted Answer

Here are the differentiations for each function using implicit differentiation:

41. Differentiate $x^2 y + y^3 = 1$

Differentiate both sides with respect to $x$ .
$\frac{d}{dx}(x^2 y) + \frac{d}{dx}(y^3) = \frac{d}{dx}(1)$
Apply the product rule for $x^2 y$ and the chain rule for $y^3$ .
$(2x \cdot y + x^2 \cdot \frac{dy}{dx}) + 3y^2 \frac{dy}{dx} = 0$
Group terms containing $\frac{dy}{dx}$ .
$x^2 \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = -2xy$
Factor out $\frac{dy}{dx}$ .
$\frac{dy}{dx}(x^2 + 3y^2) = -2xy$
Solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{-2xy}{x^2 + 3y^2}$
The derivative is $\boxed{\frac{-2xy}{x^2 + 3y^2}}$ .

42. Differentiate $\ln y + x^2 = 3$
Differentiate both sides with respect to $x$ .
$\frac{d}{dx}(\ln y) + \frac{d}{dx}(x^2) = \frac{d}{dx}(3)$
Apply the chain rule for $\ln y$ .
$\frac{1}{y} \frac{dy}{dx} + 2x = 0$
Isolate the term with $\frac{dy}{dx}$ .
$\frac{1}{y} \frac{dy}{dx} = -2x$
Solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = -2xy$
The derivative is $\boxed{-2xy}$ .

43. Differentiate $e^{xy} = x + y$
Differentiate both sides with respect to $x$ .
$\frac{d}{dx}(e^{xy}) = \frac{d}{dx}(x) + \frac{d}{dx}(y)$
Apply the chain rule and product rule for $e^{xy}$ .
$e^{xy} \cdot \frac{d}{dx}(xy) = 1 + \frac{dy}{dx}$ $e^{xy} (1 \cdot y + x \cdot \frac{dy}{dx}) = 1 + \frac{dy}{dx}$ $ye^{xy} + xe^{xy} \frac{dy}{dx} = 1 + \frac{dy}{dx}$
Group terms containing $\frac{dy}{dx}$ .
$xe^{xy} \frac{dy}{dx} - \frac{dy}{dx} = 1 - ye^{xy}$
Factor out $\frac{dy}{dx}$ .
$\frac{dy}{dx}(xe^{xy} - 1) = 1 - ye^{xy}$
Solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{1 - ye^{xy}}{xe^{xy} - 1}$
The derivative is $\boxed{\frac{1 - ye^{xy}}{xe^{xy} - 1}}$ .

44. Differentiate $x^2 + \sin y = y$
Differentiate both sides with respect to $x$ .
$\frac{d}{dx}(x^2) + \frac{d}{dx}(\sin y) = \frac{d}{dx}(y)$
Apply the chain rule for $\sin y$ .
$2x + \cos y \frac{dy}{dx} = \frac{dy}{dx}$
Group terms containing $\frac{dy}{dx}$ .
$2x = \frac{dy}{dx} - \cos y \frac{dy}{dx}$
Factor out $\frac{dy}{dx}$ .
$2x = \frac{dy}{dx}(1 - \cos y)$
Solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{2x}{1 - \cos y}$
The derivative is $\boxed{\frac{2x}{1 - \cos y}}$ .

45. Differentiate $x^3 + y^3 + 3xy = 1$
Differentiate both sides with respect to $x$ .
$\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) + \frac{d}{dx}(3xy) = \frac{d}{dx}(1)$
Apply the chain rule for $y^3$ and the product rule for $3xy$ .
$3x^2 + 3y^2 \frac{dy}{dx} + 3(1 \cdot y + x \cdot \frac{dy}{dx}) = 0$ $3x^2 + 3y^2 \frac{dy}{dx} + 3y + 3x \frac{dy}{dx} = 0$
Group terms containing $\frac{dy}{dx}$ .
$3y^2 \frac{dy}{dx} + 3x \frac{dy}{dx} = -3x^2 - 3y$
Factor out $\frac{dy}{dx}$ .
$\frac{dy}{dx}(3y^2 + 3x) = -3x^2 - 3y$
Solve for $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{-3x^2 - 3y}{3y^2 + 3x}$
Simplify by dividing the numerator and denominator by 3.
$\frac{dy}{dx} = \frac{-(x^2 + y)}{y^2 + x}$
The derivative is $\boxed{\frac{-(x^2 + y)}{y^2 + x}}$ .

Differentiate x2 y + y3 = 1, y + x2 = 3, and e^xy = x + y using implicit differentiation.

Step-by-step solution

Differentiate both sides with respect to $x$ .

Apply the product rule for $x^2 y$ and the chain rule for $y^3$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule for $\ln y$ .

Isolate the term with $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule and product rule for $e^{xy}$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule for $\sin y$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule for $y^3$ and the product rule for $3xy$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Simplify by dividing the numerator and denominator by 3.

Still stuck on this one?

More Mathematics Questions

Differentiate x2 y + y3 = 1, y + x2 = 3, and e^xy = x + y using implicit differentiation.

Step-by-step solution

Differentiate both sides with respect to xxx.

Apply the product rule for x2yx^2 yx2y and the chain rule for y3y^3y3.

Group terms containing dydx\frac{dy}{dx}dxdy​.

Factor out dydx\frac{dy}{dx}dxdy​.

Solve for dydx\frac{dy}{dx}dxdy​.

Differentiate both sides with respect to xxx.

Apply the chain rule for ln⁡y\ln ylny.

Isolate the term with dydx\frac{dy}{dx}dxdy​.

Solve for dydx\frac{dy}{dx}dxdy​.

Differentiate both sides with respect to xxx.

Apply the chain rule and product rule for exye^{xy}exy.

Group terms containing dydx\frac{dy}{dx}dxdy​.

Factor out dydx\frac{dy}{dx}dxdy​.

Solve for dydx\frac{dy}{dx}dxdy​.

Differentiate both sides with respect to xxx.

Apply the chain rule for sin⁡y\sin ysiny.

Group terms containing dydx\frac{dy}{dx}dxdy​.

Factor out dydx\frac{dy}{dx}dxdy​.

Solve for dydx\frac{dy}{dx}dxdy​.

Differentiate both sides with respect to xxx.

Apply the chain rule for y3y^3y3 and the product rule for 3xy3xy3xy.

Group terms containing dydx\frac{dy}{dx}dxdy​.

Factor out dydx\frac{dy}{dx}dxdy​.

Solve for dydx\frac{dy}{dx}dxdy​.

Simplify by dividing the numerator and denominator by 3.

Still stuck on this one?

More Mathematics Questions

Differentiate both sides with respect to $x$ .

Apply the product rule for $x^2 y$ and the chain rule for $y^3$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule for $\ln y$ .

Isolate the term with $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule and product rule for $e^{xy}$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule for $\sin y$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .

Differentiate both sides with respect to $x$ .

Apply the chain rule for $y^3$ and the product rule for $3xy$ .

Group terms containing $\frac{dy}{dx}$ .

Factor out $\frac{dy}{dx}$ .

Solve for $\frac{dy}{dx}$ .