Find dy/dx 4x^2-2xy^2+5y-9=0

Solution:

Step 1:

Take the derivative of the entire equation with respect to $x$: $\frac{d}{dx}(4x^2 - 2xy^2 + 5y - 9) = \frac{d}{dx}(0)$.

Step 2:

Differentiate each term of the equation separately.

Step 2.1:

Apply the Sum Rule to differentiate the sum of the terms: $\frac{d}{dx}(4x^2) + \frac{d}{dx}(-2xy^2) + \frac{d}{dx}(5y) + \frac{d}{dx}(-9)$.

Step 2.2:

Find the derivative of $4x^2$ with respect to $x$.

Step 2.2.1:

As $4$ is a constant, use the constant multiple rule: $4\frac{d}{dx}(x^2)$.

Step 2.2.2:

Apply the Power Rule, which states that the derivative of $x^n$ is $nx^{n-1}$: $4(2x)$.

Step 2.2.3:

Multiply $4$ by $2$: $8x$.

Step 2.3:

Find the derivative of $-2xy^2$ with respect to $x$.

Step 2.3.1:

As $-2$ is a constant, use the constant multiple rule: $-2\frac{d}{dx}(xy^2)$.

Step 2.3.2:

Apply the Product Rule, which states that the derivative of $f(x)g(x)$ is $f(x)g'(x) + g(x)f'(x)$: $-2(x\frac{d}{dx}(y^2) + y^2\frac{d}{dx}(x))$.

Step 2.3.3:

Differentiate $y^2$ using the Chain Rule, which involves taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Step 2.3.3.1:

Let $u = y$ and apply the Chain Rule: $-2(x(2u\frac{dy}{dx}) + y^2)$.

Step 2.3.3.2:

Differentiate $u^2$ using the Power Rule: $-2(x(2y\frac{dy}{dx}) + y^2)$.

Step 2.3.3.3:

Substitute $u$ back with $y$: $-2(x(2y\frac{dy}{dx}) + y^2)$.

Step 2.3.4:

Rewrite $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$.

Step 2.3.5:

Differentiate $x$ using the Power Rule: $-2(2xy\frac{dy}{dx} + y^2)$.

Step 2.3.6:

Rearrange the terms: $-4xy\frac{dy}{dx} - 2y^2$.

Step 2.3.7:

Simplify the expression: $-4xy\frac{dy}{dx} - 2y^2$.

Step 2.4:

Find the derivative of $5y$ with respect to $x$.

Step 2.4.1:

As $5$ is a constant, use the constant multiple rule: $5\frac{d}{dx}(y)$.

Step 2.4.2:

Rewrite $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$: $5\frac{dy}{dx}$.

Step 2.5:

Since $-9$ is a constant, its derivative is $0$.

Step 2.6:

Combine all the differentiated terms.

Step 2.6.1:

Apply the distributive property: $8x - 4xy\frac{dy}{dx} - 2y^2 + 5\frac{dy}{dx}$.

Step 2.6.2:

Combine like terms.

Step 2.6.2.1:

Simplify the expression: $8x - 4xy\frac{dy}{dx} - 2y^2 + 5\frac{dy}{dx}$.

Step 2.6.2.2:

Combine the terms: $8x - 4xy\frac{dy}{dx} - 2y^2 + 5\frac{dy}{dx}$.

Step 2.6.3:

Arrange the terms in a standard form: $-2y^2 - 4xy\frac{dy}{dx} + 8x + 5\frac{dy}{dx}$.

Step 3:

The derivative of the right side of the equation, $0$, is also $0$.

Step 4:

Set the left side of the derivative equation equal to the right side: $-2y^2 - 4xy\frac{dy}{dx} + 8x + 5\frac{dy}{dx} = 0$.

Step 5:

Solve for $\frac{dy}{dx}$.

Step 5.1:

Move all terms not containing $\frac{dy}{dx}$ to the other side of the equation.

Step 5.1.1:

Add $2y^2$ to both sides: $-4xy\frac{dy}{dx} + 5\frac{dy}{dx} = 2y^2 - 8x$.

Step 5.1.2:

Subtract $8x$ from both sides: $-4xy\frac{dy}{dx} + 5\frac{dy}{dx} = 2y^2 - 8x$.

Step 5.2:

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

Step 5.2.1:

Factor $\frac{dy}{dx}$ out of $-4xy\frac{dy}{dx}$: $\frac{dy}{dx}(-4xy) + 5\frac{dy}{dx} = 2y^2 - 8x$.

Step 5.2.2:

Factor $\frac{dy}{dx}$ out of $5\frac{dy}{dx}$: $\frac{dy}{dx}(-4xy + 5) = 2y^2 - 8x$.

Step 5.2.3:

Factor $\frac{dy}{dx}$ out: $\frac{dy}{dx}(-4xy + 5) = 2y^2 - 8x$.

Step 5.3:

Divide both sides by $-4xy + 5$ and simplify.

Step 5.3.1:

Divide both sides: $\frac{dy}{dx} = \frac{2y^2 - 8x}{-4xy + 5}$.

Step 5.3.2:

Simplify the expression: $\frac{dy}{dx} = \frac{2y^2 - 8x}{-4xy + 5}$.

Step 6:

Replace $\frac{dy}{dx}$ with the simplified expression: $\frac{dy}{dx} = \frac{2y^2 - 8x}{-4xy + 5}$.

Knowledge Notes:

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

2. Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

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

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

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

6. Implicit Differentiation: When a function is not given explicitly as $y=f(x)$, but rather implicitly as in $F(x,y)=0$, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.