Find dy/dx 5e^(xy)+3y^2=7

Solution:

Step 1:

Apply the derivative operator to both sides of the given equation: $\frac{d}{dx}(5e^{xy} + 3y^2) = \frac{d}{dx}(7)$.

Step 2:

Take the derivative of the left-hand side term by term.

Step 2.1:

Use the Sum Rule to separate the derivatives: $\frac{d}{dx}(5e^{xy}) + \frac{d}{dx}(3y^2)$.

Step 2.2:

Find the derivative of $5e^{xy}$ with respect to $x$.

Step 2.2.1:

Treat the constant 5 as a multiplier: $5\frac{d}{dx}(e^{xy})$.

Step 2.2.2:

Apply the Chain Rule to $e^{xy}$, considering $e^u$ where $u = xy$.

Step 2.2.2.1:

Set $u_1 = xy$ and differentiate: $5\left(\frac{d}{du_1}(e^{u_1})\frac{d}{dx}(xy)\right)$.

Step 2.2.2.2:

Utilize the Exponential Rule: $5\left(e^{u_1}\frac{d}{dx}(xy)\right)$, where $e^{u_1}$ differentiates to $e^{u_1}\ln(e)$.

Step 2.2.2.3:

Substitute $u_1$ back with $xy$: $5\left(e^{xy}\frac{d}{dx}(xy)\right)$.

Step 2.2.3:

Differentiate $xy$ using the Product Rule: $5\left(e^{xy}(x\frac{d}{dx}(y) + y\frac{d}{dx}(x))\right)$.

Step 2.2.4:

Recognize that $\frac{d}{dx}(y)$ is $\frac{dy}{dx}$.

Step 2.2.5:

Apply the Power Rule to $x$: $5\left(e^{xy}(xy + y\cdot 1)\right)$.

Step 2.2.6:

Simplify the expression: $5e^{xy}(xy + y)$.

Step 2.3:

Now, differentiate $3y^2$ with respect to $x$.

Step 2.3.1:

Extract the constant 3: $3\frac{d}{dx}(y^2)$.

Step 2.3.2:

Apply the Chain Rule to $y^2$, treating it as $(u_2)^2$ where $u_2 = y$.

Step 2.3.2.1:

Set $u_2 = y$ and differentiate: $3\left(\frac{d}{du_2}(u_2^2)\frac{d}{dx}(y)\right)$.

Step 2.3.2.2:

Use the Power Rule on $u_2^2$: $3\left(2u_2\frac{d}{dx}(y)\right)$.

Step 2.3.2.3:

Replace $u_2$ with $y$: $3\left(2y\frac{d}{dx}(y)\right)$.

Step 2.3.3:

Recognize that $\frac{d}{dx}(y)$ is $\frac{dy}{dx}$.

Step 2.3.4:

Combine the constants: $6y\frac{dy}{dx}$.

Step 2.4:

Combine the differentiated terms.

Step 2.4.1:

Distribute $5e^{xy}$: $5e^{xy}xy + 5e^{xy}y + 6y\frac{dy}{dx}$.

Step 2.4.2:

Arrange the terms: $5e^{xy}xy + 5e^{xy}y + 6y\frac{dy}{dx}$.

Step 3:

Differentiate the constant 7 with respect to $x$: $0$.

Step 4:

Set the differentiated left-hand side equal to the differentiated right-hand side: $5e^{xy}xy + 5e^{xy}y + 6y\frac{dy}{dx} = 0$.

Step 5:

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

Step 5.1:

Rearrange the terms: $5xye^{xy} + 5ye^{xy} + 6y\frac{dy}{dx} = 0$.

Step 5.2:

Move $5ye^{xy}$ to the other side: $5xye^{xy} + 6y\frac{dy}{dx} = -5ye^{xy}$.

Step 5.3:

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

Step 5.3.1:

Factor $y$ from $5xye^{xy}$: $y(5xe^{xy}) + 6y\frac{dy}{dx} = -5ye^{xy}$.

Step 5.3.2:

Factor $y$ from $6y\frac{dy}{dx}$: $y(5xe^{xy}) + y(6\frac{dy}{dx}) = -5ye^{xy}$.

Step 5.3.3:

Factor $y$ from the entire left-hand side: $y(5xe^{xy} + 6\frac{dy}{dx}) = -5ye^{xy}$.

Step 5.4:

Divide both sides by $5xe^{xy} + 6y$ to isolate $\frac{dy}{dx}$.

Step 5.4.1:

Divide the equation: $\frac{y(5xe^{xy} + 6y)}{5xe^{xy} + 6y} = \frac{-5ye^{xy}}{5xe^{xy} + 6y}$.

Step 5.4.2:

Simplify the left-hand side by canceling out common factors.

Step 5.4.2.1:

Cancel the common factor: $\frac{y(\cancel{5xe^{xy} + 6y})}{\cancel{5xe^{xy} + 6y}} = \frac{-5ye^{xy}}{5xe^{xy} + 6y}$.

Step 5.4.2.1.2:

Simplify the division: $y = \frac{-5ye^{xy}}{5xe^{xy} + 6y}$.

Step 5.4.3:

Simplify the right-hand side by moving the negative sign in front of the fraction.

Step 6:

Substitute $\frac{dy}{dx}$ for $y$ in the final expression: $\frac{dy}{dx} = -\frac{5ye^{xy}}{5xe^{xy} + 6y}$.

Knowledge Notes:

1. Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

2. Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

3. Product Rule: The derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

4. Exponential Rule: The derivative of $e^u$ where $u$ is a function of $x$ is $e^u$ times the derivative of $u$.

5. Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

6. Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.

7. Differentiation of Constants: The derivative of a constant with respect to $x$ is zero.

8. Implicit Differentiation: When a function is not given in the form $y=f(x)$, differentiation is performed with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule as necessary.