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

Solution:

Step 1

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

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 $y^{\prime }$: $e^x-y^{\prime }$.

Step 3

Differentiate the right-hand side:

Step 3.1

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

Step 3.2

Find the derivative of $x{y}^3$:

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: $x(3y^2{y}^{\prime })+y^3$.

Step 3.2.2.3

Substitute $u$ back with $y$: $x(3y^2{y}^{\prime })+y^3$.

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: $y^3+3x{y}^2{y}^{\prime }$.

Step 4

Combine the differentiated left and right sides: $e^x-y^{\prime }=y^3+3x{y}^2{y}^{\prime }$.

Step 5

Solve for $y^{\prime }$:

Step 5.1

Isolate terms involving $y^{\prime }$: $e^x-y^3=y^{\prime }+3x{y}^2{y}^{\prime }$.

Step 5.2

Factor out $y^{\prime }$: $y^{\prime }(1+3x{y}^2)=e^x-y^3$.

Step 5.3

Divide both sides by $(1+3x{y}^2)$ to solve for $y^{\prime }$.

Step 5.4

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

Step 6

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

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 $f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$.

4. Chain Rule: The derivative of a composite function $f(g(x))$ is $f^{\prime }(g(x))g^{\prime }(x)$.

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

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 $y^{\prime }$ and $\frac{dy}{dx}$ both represent the derivative of $y$ with respect to $x$.