Find dy/dx e^x-y=xy^3+e^2-18

Solution:

Step 1

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(e^x - y) = \frac{d}{dx}(xy^3 + e^2 - 18)$.

Step 2

Differentiate the left-hand side:

Step 2.1

Apply the Sum Rule: $\frac{d}{dx}(e^x) + \frac{d}{dx}(-y)$.

Step 2.2

Use the Exponential Rule: $e^x \ln(e) + \frac{d}{dx}(-y)$, noting that $\ln(e) = 1$.

Step 2.3

Find the derivative of $-y$:

Step 2.3.1

Since $-1$ is a constant, $\frac{d}{dx}(-y) = -\frac{d}{dx}(y)$.

Step 2.3.2

Represent $\frac{d}{dx}(y)$ as $y'$: $e^x - y'$.

Step 3

Differentiate the right-hand side:

Step 3.1

Use the Sum Rule: $\frac{d}{dx}(xy^3) + \frac{d}{dx}(e^2) + \frac{d}{dx}(-18)$.

Step 3.2

Find the derivative of $xy^3$:

Step 3.2.1

Apply the Product Rule: $y^3 \frac{d}{dx}(x) + x \frac{d}{dx}(y^3)$.

Step 3.2.2

Use the Chain Rule for $y^3$:

Step 3.2.2.1

Let $u = y$: $x(3u^2 \frac{d}{dx}(y)) + y^3$.

Step 3.2.2.2

Apply the Power Rule: $x(3y^2 y') + y^3$.

Step 3.2.2.3

Substitute $u$ back with $y$: $x(3y^2 y') + y^3$.

Step 3.3

Differentiate the constants:

Step 3.3.1

Since $e^2$ is constant, its derivative is $0$.

Step 3.3.2

The derivative of $-18$ is also $0$.

Step 3.4

Simplify the expression: $y^3 + 3xy^2 y'$.

Step 4

Combine the differentiated left and right sides: $e^x - y' = y^3 + 3xy^2 y'$.

Step 5

Solve for $y'$:

Step 5.1

Isolate terms involving $y'$: $e^x - y^3 = y' + 3xy^2 y'$.

Step 5.2

Factor out $y'$: $y'(1 + 3xy^2) = e^x - y^3$.

Step 5.3

Divide both sides by $(1 + 3xy^2)$ to solve for $y'$.

Step 5.4

Simplify the expression for $y'$: $y' = \frac{e^x - y^3}{1 + 3xy^2}$.

Step 6

Replace $y'$ with $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{e^x - y^3}{1 + 3xy^2}$.

Knowledge Notes:

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

2. Exponential Rule: The derivative of $e^x$ is $e^x$. More generally, the derivative of $a^x$ is $a^x \ln(a)$.

3. Product Rule: The derivative of a product of two functions $f(x)g(x)$ is given by $f'(x)g(x) + f(x)g'(x)$.

4. Chain Rule: The derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$.

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

6. Constant Rule: The derivative of a constant is zero.

7. Implicit Differentiation: When a function is not given in the form $y=f(x)$, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.

8. Notation: The notation $y'$ and $\frac{dy}{dx}$ both represent the derivative of $y$ with respect to $x$.