Find dy/dx xe^y=ysin(x)

Solution:

Step 1:

Take the derivative of each side with respect to $x$: $\frac{d}{dx}(xe^y) = \frac{d}{dx}(y\sin(x))$.

Step 2:

Apply the derivative to the left-hand side.

Step 2.1:

Invoke the Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = x$ and $v = e^y$. This gives $x\frac{d}{dx}(e^y) + e^y\frac{d}{dx}(x)$.

Step 2.2:

Apply the Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$, where $f(u) = e^u$ and $g(x) = y$.

Step 2.2.1:

Set $u = y$ to prepare for the Chain Rule: $x(\frac{d}{du}(e^u)\frac{dx}{du}) + e^y\frac{d}{dx}(x)$.

Step 2.2.2:

Apply the Exponential Rule: $\frac{d}{du}(a^u) = a^u\ln(a)$, where $a = e$. This results in $x(e^u\frac{dx}{du}) + e^y\frac{d}{dx}(x)$.

Step 2.2.3:

Replace $u$ with $y$: $x(e^y\frac{dx}{du}) + e^y\frac{d}{dx}(x)$.

Step 2.3:

Express $\frac{dx}{du}$ as $\frac{dy}{dx}$: $xe^y\frac{dy}{dx} + e^y\frac{d}{dx}(x)$.

Step 2.4:

Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$: $xe^y\frac{dy}{dx} + e^y\cdot 1$.

Step 2.5:

Multiply $e^y$ by $1$: $xe^y\frac{dy}{dx} + e^y$.

Step 3:

Differentiate the right-hand side.

Step 3.1:

Use the Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = y$ and $v = \sin(x)$. This gives $y\frac{d}{dx}(\sin(x)) + \sin(x)\frac{d}{dx}(y)$.

Step 3.2:

The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$: $y\cos(x) + \sin(x)\frac{d}{dx}(y)$.

Step 3.3:

Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$: $y\cos(x) + \sin(x)\frac{dy}{dx}$.

Step 4:

Combine the derivatives from both sides: $xe^y\frac{dy}{dx} + e^y = y\cos(x) + \sin(x)\frac{dy}{dx}$.

Step 5:

Isolate $\frac{dy}{dx}$.

Step 5.1:

Simplify the left-hand side.

Step 5.1.1:

Rearrange the terms: $xye^y + e^y = y\cos(x) + \sin(x)y$.

Step 5.2:

Simplify the right-hand side.

Step 5.2.1:

Rearrange the terms: $xye^y + e^y = y\cos(x) + y\sin(x)$.

Step 5.3:

Subtract $y\sin(x)$ from both sides: $xye^y + e^y - y\sin(x) = y\cos(x)$.

Step 5.4:

Subtract $e^y$ from both sides: $xye^y - y\sin(x) = y\cos(x) - e^y$.

Step 5.5:

Factor out $y$ from the left-hand side.

Step 5.5.1:

Factor $y$ from $xye^y$: $y(xe^y) - y\sin(x) = y\cos(x) - e^y$.

Step 5.5.2:

Factor $y$ from $-y\sin(x)$: $y(xe^y) - y\sin(x) = y\cos(x) - e^y$.

Step 5.5.3:

Factor $y$ from the entire expression: $y(xe^y - \sin(x)) = y\cos(x) - e^y$.

Step 5.6:

Rewrite $-y\sin(x)$ as $-y\sin(x)$: $y(xe^y - \sin(x)) = y\cos(x) - e^y$.

Step 5.7:

Divide by $xe^y - \sin(x)$ and simplify.

Step 5.7.1:

Divide each term: $\frac{y(xe^y - \sin(x))}{xe^y - \sin(x)} = \frac{y\cos(x)}{xe^y - \sin(x)} - \frac{e^y}{xe^y - \sin(x)}$.

Step 5.7.2:

Simplify the left side by canceling out common factors.

Step 5.7.2.1:

Cancel the common factor: $\frac{y}{1} = \frac{y\cos(x)}{xe^y - \sin(x)} - \frac{e^y}{xe^y - \sin(x)}$.

Step 5.7.3:

Simplify the right side by combining terms over a common denominator: $y = \frac{y\cos(x) - e^y}{xe^y - \sin(x)}$.

Step 6:

Replace $y$ with $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{y\cos(x) - e^y}{xe^y - \sin(x)}$.

Knowledge Notes:

1. Product Rule: Used when differentiating a product of two functions, $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$.

2. Chain Rule: Used when differentiating a composite function, $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$.

3. Exponential Rule: When differentiating an exponential function, $\frac{d}{dx}(a^x) = a^x\ln(a)$.

4. Power Rule: Used when differentiating a power of $x$, $\frac{d}{dx}(x^n) = nx^{n-1}$.

5. Simplifying Expressions: Involves factoring, canceling common factors, and combining terms over a common denominator to simplify the expression.

6. Implicit Differentiation: Used when a function cannot be explicitly solved for one variable in terms of another, which is the case in this problem where $y$ is a function of $x$ but is not isolated on one side of the equation.