Find dy/dx e^(4xy)=3y

Solution:

Step 1:

Apply the derivative operator $\frac{d}{dx}$ to both sides of the equation $e^{4xy} = 3y$.

Step 2:

Take the derivative of the left-hand side.

Step 2.1:

Utilize the chain rule for differentiation, which is $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = e^x$ and $g(x) = 4xy$.

Step 2.1.1:

Introduce $u = 4xy$ and differentiate $e^u$ with respect to $u$ and $4xy$ with respect to $x$: $\frac{du}{dx}e^u$.

Step 2.1.2:

Apply the rule for differentiating exponential functions, $\frac{d}{du}[e^u] = e^u \ln(e)$, since $a = e$: $e^u \frac{du}{dx}$.

Step 2.1.3:

Substitute back $u = 4xy$: $e^{4xy} \frac{d}{dx}[4xy]$.

Step 2.2:

Recognize that $4$ is a constant and differentiate $4xy$ with respect to $x$: $e^{4xy}(4 \frac{d}{dx}[xy])$.

Step 2.3:

Apply the product rule for differentiation, $\frac{d}{dx}[fg] = f \frac{dg}{dx} + g \frac{df}{dx}$, where $f(x) = x$ and $g(x) = y$: $e^{4xy}(4(x \frac{dy}{dx} + y))$.

Step 2.4:

Express $\frac{d}{dx}[y]$ as $\frac{dy}{dx}$: $e^{4xy}(4(xy + y))$.

Step 2.5:

Differentiate $x$ using the power rule, $\frac{d}{dx}[x^n] = nx^{n-1}$, where $n = 1$: $e^{4xy}(4(xy + y \cdot 1))$.

Step 2.6:

Multiply $y$ by $1$: $e^{4xy}(4(xy + y))$.

Step 2.7:

Simplify the expression.

Step 2.7.1:

Distribute $4$ across the terms: $e^{4xy}(4xy + 4y)$.

Step 2.7.2:

Separate the terms: $4e^{4xy}xy + 4e^{4xy}y$.

Step 2.7.3:

Arrange the terms in a standard form: $4e^{4xy}xy + 4e^{4xy}y$.

Step 3:

Differentiate the right-hand side of the equation.

Step 3.1:

Since $3$ is a constant, differentiate $3y$ with respect to $x$: $3\frac{dy}{dx}$.

Step 3.2:

Represent $\frac{dy}{dx}$ as $\frac{dy}{dx}$: $3\frac{dy}{dx}$.

Step 4:

Combine the derivatives from the left and right sides to form an equation: $4e^{4xy}xy + 4e^{4xy}y = 3\frac{dy}{dx}$.

Step 5:

Isolate $\frac{dy}{dx}$ to solve for it.

Step 5.1:

Rearrange the terms: $4xye^{4xy} + 4ye^{4xy} = 3\frac{dy}{dx}$.

Step 5.2:

Subtract $3\frac{dy}{dx}$ from both sides: $4xye^{4xy} + 4ye^{4xy} - 3\frac{dy}{dx} = 0$.

Step 5.3:

Subtract $4ye^{4xy}$ from both sides: $4xye^{4xy} - 3\frac{dy}{dx} = -4ye^{4xy}$.

Step 5.4:

Factor out $\frac{dy}{dx}$ from the left-hand side.

Step 5.4.1:

Factor out $\frac{dy}{dx}$ from $4xye^{4xy}$: $\frac{dy}{dx}(4xe^{4xy}) - 3\frac{dy}{dx} = -4ye^{4xy}$.

Step 5.4.2:

Factor out $\frac{dy}{dx}$ from $-3\frac{dy}{dx}$: $\frac{dy}{dx}(4xe^{4xy} - 3) = -4ye^{4xy}$.

Step 5.4.3:

Combine the terms: $\frac{dy}{dx}(4xe^{4xy} - 3) = -4ye^{4xy}$.

Step 5.5:

Divide both sides by $(4xe^{4xy} - 3)$ to isolate $\frac{dy}{dx}$.

Step 5.5.1:

Divide both sides: $\frac{\frac{dy}{dx}(4xe^{4xy} - 3)}{4xe^{4xy} - 3} = \frac{-4ye^{4xy}}{4xe^{4xy} - 3}$.

Step 5.5.2:

Simplify the left side by canceling out the common factor.

Step 5.5.2.1:

Cancel the common factors: $\frac{\cancel{(4xe^{4xy} - 3)}}{\cancel{(4xe^{4xy} - 3)}}\frac{dy}{dx} = \frac{-4ye^{4xy}}{4xe^{4xy} - 3}$.

Step 5.5.2.1.2:

Simplify the division: $\frac{dy}{dx} = \frac{-4ye^{4xy}}{4xe^{4xy} - 3}$.

Step 5.5.3:

Simplify the right side by moving the negative sign in front of the fraction: $\frac{dy}{dx} = -\frac{4ye^{4xy}}{4xe^{4xy} - 3}$.

Step 6:

Finalize the expression for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{4ye^{4xy}}{4xe^{4xy} - 3}$.

Knowledge Notes:

1. Chain Rule: This is a fundamental rule in calculus used to differentiate compositions of functions. If $u = g(x)$ and $y = f(u)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

2. Product Rule: This rule is used to differentiate products of two functions $f(x)$ and $g(x)$. The derivative is $\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$.

3. Exponential Rule: The derivative of an exponential function $a^u$ with respect to $u$ is $a^u \ln(a)$, which simplifies to $e^u$ when $a = e$.

4. Power Rule: For any real number $n$, the power rule states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

5. Simplification: This involves algebraic manipulation such as distributing factors and combining like terms to simplify expressions.

6. Factoring: This is the process of writing an expression as a product of its factors. It is often used to simplify equations and solve for variables.

7. Isolating Variables: In solving equations, it is often necessary to isolate the variable of interest on one side of the equation to find its value.