Find dy/dx y^3-xy=-6

Solution:

Step:1

Apply differentiation to both sides of the given equation with respect to $x$: $\frac{d}{dx}(y^3 - xy) = \frac{d}{dx}(-6)$.

Step:2

Perform differentiation on the left-hand side term by term.

Step:2.1

Using the Sum Rule, differentiate $y^3$ and $-xy$ separately: $\frac{d}{dx}(y^3) + \frac{d}{dx}(-xy)$.

Step:2.2

Find the derivative of $y^3$ with respect to $x$.

Step:2.2.1

Apply the Chain Rule, where $f(x) = x^3$ and $g(x) = y$, to differentiate $y^3$: $\frac{d}{du}(u^3)\frac{dy}{dx}$, with $u = y$.

Step:2.2.1.1

Substitute $u$ for $y$ to prepare for the Chain Rule: $\frac{d}{du}(u^3)\frac{dy}{dx} + \frac{d}{dx}(-xy)$.

Step:2.2.1.2

Differentiate $u^3$ using the Power Rule, where $n = 3$: $3u^2\frac{dy}{dx} + \frac{d}{dx}(-xy)$.

Step:2.2.1.3

Replace $u$ back with $y$: $3y^2\frac{dy}{dx} + \frac{d}{dx}(-xy)$.

Step:2.2.2

Express $\frac{dy}{dx}$ as $y'$: $3y^2y' + \frac{d}{dx}(-xy)$.

Step:2.3

Differentiate $-xy$ with respect to $x$.

Step:2.3.1

Treat $-1$ as a constant multiplier and differentiate $-xy$: $3y^2y' - \frac{d}{dx}(xy)$.

Step:2.3.2

Apply the Product Rule to differentiate $xy$, where $f(x) = x$ and $g(x) = y$: $3y^2y' - (x\frac{dy}{dx} + y\frac{dx}{dx})$.

Step:2.3.3

Replace $\frac{dy}{dx}$ with $y'$: $3y^2y' - (xy' + y)$.

Step:2.3.4

Use the Power Rule to differentiate $x$ with $n = 1$: $3y^2y' - (xy' + y\cdot1)$.

Step:2.3.5

Multiply $y$ by $1$: $3y^2y' - (xy' + y)$.

Step:2.4

Combine and simplify the terms.

Step:2.4.1

Distribute $y'$: $3y^3y' - xy' - y$.

Step:2.4.2

Remove any unnecessary brackets: $3y^2y' - xy' - y$.

Step:3

Differentiate the constant $-6$ with respect to $x$: $0$.

Step:4

Combine the differentiated left side with the right side: $3y^2y' - xy' - y = 0$.

Step:5

Isolate $y'$ (the derivative of $y$ with respect to $x$).

Step:5.1

Add $xy'$ and $y$ to both sides: $3y^2y' = xy' + y$.

Step:5.2

Factor out $y'$ from the left side.

Step:5.2.1

Factor $y'$ from $3y^2y'$: $y'(3y^2) - xy' = y$.

Step:5.2.2

Factor $y'$ from $-xy'$: $y'(3y^2) - y'(x) = y$.

Step:5.2.3

Combine the terms with $y'$: $y'(3y^2 - x) = y$.

Step:5.3

Divide both sides by $(3y^2 - x)$ to solve for $y'$.

Step:5.3.1

Divide both sides by $(3y^2 - x)$: $\frac{y'(3y^2 - x)}{3y^2 - x} = \frac{y}{3y^2 - x}$.

Step:5.3.2

Simplify the left side.

Step:5.3.2.1

Cancel out the common factor $(3y^2 - x)$: $\frac{y'(\cancel{3y^2 - x})}{\cancel{3y^2 - x}} = \frac{y}{3y^2 - x}$.

Step:5.3.2.1.1

Complete the cancellation: $y' = \frac{y}{3y^2 - x}$.

Step:6

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

Knowledge Notes:

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

2. Chain Rule: Used to differentiate the composition of functions. It states that if $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is $f'(g(x))g'(x)$.

3. Power Rule: A basic differentiation rule that states if $f(x) = x^n$, then the derivative of $f$ with respect to $x$ is $f'(x) = nx^{n-1}$.

4. Product Rule: Used to differentiate the product of two functions. If $u(x)$ and $v(x)$ are functions of $x$, then the derivative of their product $u(x)v(x)$ is given by $u'(x)v(x) + u(x)v'(x)$.

5. Differentiation of Constants: The derivative of a constant is zero.

6. Factoring: A method used to simplify expressions and solve equations by expressing a polynomial as the product of its factors.

7. Solving for a Variable: The process of isolating a specific variable on one side of an equation to find its value in terms of other variables.