Find dy/dx 6x^3+5y^3=11xy

Solution:

Step 1:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(6x^3+5y^3)=\frac{d}{dx}(11xy)$.

Step 2:

Derive the left-hand side of the equation.

Step 2.1:

Apply the Sum Rule to differentiate $6x^3+5y^3$ as the sum of derivatives: $\frac{d}{dx}(6x^3)+\frac{d}{dx}(5y^3)$.

Step 2.2:

Find the derivative of $6x^3$.

Step 2.2.1:

As $6$ is a constant, use the constant multiplier rule: $6\frac{d}{dx}(x^3)$.

Step 2.2.2:

Apply the Power Rule, where the derivative of $x^n$ is $n{x}^{n-1}$ for $n=3$: $6(3x^2)$.

Step 2.2.3:

Simplify the multiplication: $18x^2$.

Step 2.3:

Find the derivative of $5y^3$.

Step 2.3.1:

As $5$ is a constant, use the constant multiplier rule: $5\frac{d}{dx}(y^3)$.

Step 2.3.2:

Use the Chain Rule for the derivative of a composite function $f(g(x))$: $f^{\prime }(g(x))g^{\prime }(x)$ with $f(x)=x^3$ and $g(x)=y$.

Step 2.3.2.1:

Introduce $u=y$ to apply the Chain Rule: $5(\frac{d}{du}(u^3)\frac{d}{dx}(y))$.

Step 2.3.2.2:

Apply the Power Rule to $u^3$: $5(3u^2\frac{d}{dx}(y))$.

Step 2.3.2.3:

Substitute back $u=y$: $5(3y^2\frac{d}{dx}(y))$.

Step 2.3.3:

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

Step 2.3.4:

Simplify the constants: $15y^2\frac{dy}{dx}$.

Step 3:

Derive the right-hand side of the equation.

Step 3.1:

As $11$ is a constant, use the constant multiplier rule: $11\frac{d}{dx}(xy)$.

Step 3.2:

Apply the Product Rule for the derivative of the product of two functions $f(x)g(x)$: $f(x)g^{\prime }(x)+g(x)f^{\prime }(x)$ with $f(x)=x$ and $g(x)=y$.

Step 3.3:

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

Step 3.4:

Apply the Power Rule to $x$: $11(xy+y(1))$.

Step 3.5:

Simplify the multiplication: $11y$.

Step 3.6:

Distribute $11$ over $xy+y$: $11xy+11y$.

Step 4:

Combine the derived parts to form an equation: $18x^2+15y^2\frac{dy}{dx}=11xy+11y$.

Step 5:

Isolate $\frac{dy}{dx}$.

Step 5.1:

Subtract $11xy$ from both sides: $18x^2+15y^2\frac{dy}{dx}-11xy=11y$.

Step 5.2:

Subtract $18x^2$ from both sides: $15y^2\frac{dy}{dx}-11xy=11y-18x^2$.

Step 5.3:

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

Step 5.3.1:

Factor $\frac{dy}{dx}$ out of $15y^2\frac{dy}{dx}$: $\frac{dy}{dx}(15y^2)-11xy=11y-18x^2$.

Step 5.3.2:

Factor $\frac{dy}{dx}$ out of $-11xy$: $\frac{dy}{dx}(15y^2-11x)=11y-18x^2$.

Step 5.4:

Divide each term by $(15y^2-11x)$ and simplify.

Step 5.4.1:

Divide the equation by $(15y^2-11x)$: $\frac{\frac{dy}{dx}(15y^2-11x)}{15y^2-11x}=\frac{11y-18x^2}{15y^2-11x}$.

Step 5.4.2:

Simplify the left side by canceling out the common factors.

Step 5.4.2.1:

Cancel the common factor: $\frac{dy}{dx}=\frac{11y-18x^2}{15y^2-11x}$.

Step 6:

Substitute $\frac{dy}{dx}$ with the simplified expression: $\frac{dy}{dx}=\frac{11y-18x^2}{15y^2-11x}$.

Knowledge Notes:

The problem involves finding the derivative of an implicitly defined function. The steps taken in the solution involve the application of various rules of differentiation:

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

2. Constant Multiplier 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 $n{x}^{n-1}$.

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

5. Product Rule: The derivative of the product of two functions $f(x)g(x)$ is $f(x)g^{\prime }(x)+g(x)f^{\prime }(x)$.

In the context of implicit differentiation, when differentiating terms involving $y$, it is necessary to apply the chain rule because $y$ is a function of $x$. This means that the derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$, which must be included in the differentiation process.

The final result gives the derivative $\frac{dy}{dx}$ in terms of $x$ and $y$, which is the slope of the tangent to the curve defined by the given equation at any point $(x,y)$.