Find dy/dx x^5+y^5=30xy

Solution:

Step 1

Apply differentiation to both sides of the given equation with respect to $x$: $\frac{d}{dx}(x^5+y^5)=\frac{d}{dx}(30xy)$.

Step 2

Differentiate the left-hand side term by term.

Step 2.1

Use the Sum Rule of differentiation: $\frac{d}{dx}(x^5)+\frac{d}{dx}(y^5)$.

Step 2.1.1

Apply the Power Rule to $x^5$, resulting in $5x^4$, and then differentiate $y^5$ with respect to $x$: $5x^4+\frac{d}{dx}(y^5)$.

Step 2.2

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

Step 2.2.1

Use the Chain Rule, setting $f(x)=x^5$ and $g(x)=y$: $5x^4+5y^4\frac{dy}{dx}$.

Step 2.2.2

Express $\frac{d}{dx}(y)$ as $\frac{dy}{dx}$: $5x^4+5y^4\frac{dy}{dx}$.

Step 3

Differentiate the right-hand side using the Product Rule.

Step 3.1

The derivative of $30xy$ with respect to $x$ is $30\frac{d}{dx}(xy)$.

Step 3.2

Apply the Product Rule: $30(x\frac{dy}{dx}+y\frac{d}{dx}(x))$.

Step 3.3

Substitute $\frac{d}{dx}(x)$ with $1$: $30(xy+y)$.

Step 3.4

Simplify the expression: $30xy+30y$.

Step 4

Combine the differentiated left and right sides: $5x^4+5y^4\frac{dy}{dx}=30xy+30y$.

Step 5

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

Step 5.1

Subtract $30xy$ from both sides: $5x^4+5y^4\frac{dy}{dx}-30xy=30y$.

Step 5.2

Subtract $5x^4$ from both sides: $5y^4\frac{dy}{dx}-30xy=30y-5x^4$.

Step 5.3

Factor out $5y$ from the left side.

Step 5.4

Divide both sides by $5y^4-6x$ to isolate $\frac{dy}{dx}$.

Step 6

Replace $y$ with $\frac{dy}{dx}$: $\frac{dy}{dx}=\frac{6y-x^4}{y^4-6x}$.

Knowledge Notes:

The problem-solving process involves differentiating an implicit function, where $y$ is a function of $x$, but not explicitly solved for $y$. The steps involve:

1. Differentiation Rules: The Sum Rule allows the differentiation of each term separately. The Power Rule states that the derivative of $x^n$ is $n{x}^{n-1}$.

2. Chain Rule: This rule is used when differentiating a composite function $f(g(x))$. It states that the derivative is $f^{\prime }(g(x))g^{\prime }(x)$.

3. Product Rule: When differentiating a product of two functions, $f(x)g(x)$, the derivative is $f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$.

4. Implicit Differentiation: This technique is used when a function is not given explicitly. Instead of solving for $y$ and then differentiating, we differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.

5. Algebraic Manipulation: After differentiating, algebraic manipulation is often necessary to isolate the derivative $\frac{dy}{dx}$.

In this problem, we used all these rules to differentiate the given equation and solve for $\frac{dy}{dx}$.