Find dy/dx x^(2/5)+y^(2/5)=1

Solution:

Step:1

Apply differentiation to both sides of the given equation with respect to $x$: $\frac{d}{dx}(x^{\frac{2}{5}} + y^{\frac{2}{5}}) = \frac{d}{dx}(1)$.

Step:2

Differentiate the terms on the left-hand side individually.

Step:2.1

Use the Sum Rule to separate the derivatives: $\frac{d}{dx}(x^{\frac{2}{5}}) + \frac{d}{dx}(y^{\frac{2}{5}})$.

Step:2.2

Find the derivative of $x^{\frac{2}{5}}$ with respect to $x$.

Step:2.2.1

Use the Power Rule, which gives the derivative of $x^n$ as $nx^{n-1}$, where $n = \frac{2}{5}$: $\frac{2}{5}x^{\frac{2}{5}-1} + \frac{d}{dx}(y^{\frac{2}{5}})$.

Step:2.2.2

Express $-1$ as a fraction with a denominator of $5$: $\frac{2}{5}x^{\frac{2}{5}-\frac{5}{5}} + \frac{d}{dx}(y^{\frac{2}{5}})$.

Step:2.2.3

Combine the exponents: $\frac{2}{5}x^{\frac{2-5}{5}} + \frac{d}{dx}(y^{\frac{2}{5}})$.

Step:2.2.4

Simplify the exponent to get: $\frac{2}{5}x^{-\frac{3}{5}} + \frac{d}{dx}(y^{\frac{2}{5}})$.

Step:2.3

Now, differentiate $y^{\frac{2}{5}}$ with respect to $x$.

Step:2.3.1

Apply the Chain Rule, where the derivative of $f(g(x))$ is $f'(g(x))g'(x)$. Here, $f(u) = u^{\frac{2}{5}}$ and $g(x) = y$.

Step:2.3.1.1

Set $u = y$: $\frac{2}{5}x^{-\frac{3}{5}} + \frac{d}{du}(u^{\frac{2}{5}})\frac{dy}{dx}$.

Step:2.3.1.2

Differentiate $u^{\frac{2}{5}}$ with respect to $u$ using the Power Rule: $\frac{2}{5}x^{-\frac{3}{5}} + \frac{2}{5}u^{\frac{2}{5}-1}\frac{dy}{dx}$.

Step:2.3.1.3

Replace $u$ back with $y$: $\frac{2}{5}x^{-\frac{3}{5}} + \frac{2}{5}y^{\frac{2}{5}-1}\frac{dy}{dx}$.

Step:2.3.2

Recognize that $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$: $\frac{2}{5}x^{-\frac{3}{5}} + \frac{2}{5}y^{-\frac{3}{5}}\frac{dy}{dx}$.

Step:3

The derivative of a constant is zero: $0$.

Step:4

Combine the derivatives to form the equation: $\frac{2}{5}x^{-\frac{3}{5}} + \frac{2}{5}y^{-\frac{3}{5}}\frac{dy}{dx} = 0$.

Step:5

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

Step:5.1

Isolate the term with $\frac{dy}{dx}$: $\frac{2}{5}y^{-\frac{3}{5}}\frac{dy}{dx} = -\frac{2}{5}x^{-\frac{3}{5}}$.

Step:5.2

Multiply both sides by $5y^{\frac{3}{5}}$ to eliminate the fraction: $2\frac{dy}{dx} = -\frac{2y^{\frac{3}{5}}}{x^{\frac{3}{5}}}$.

Step:5.3

Divide both sides by $2$ to solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{y^{\frac{3}{5}}}{x^{\frac{3}{5}}}$.

Step:6

The final derivative is $\frac{dy}{dx} = -\frac{y^{\frac{3}{5}}}{x^{\frac{3}{5}}}$.

Knowledge Notes:

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

2. Power Rule: Provides a method to differentiate functions of the form $x^n$, where the derivative is $nx^{n-1}$.

3. Chain Rule: A technique for differentiating the composition of two or more functions. If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx}$.

4. Negative Exponent Rule: For any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$.

5. Differentiating Constants: The derivative of a constant with respect to any variable is zero.

6. Solving for Derivatives: Involves isolating the derivative term on one side of the equation and simplifying the other terms to solve for the derivative.

7. Latex Formatting: Mathematical expressions are formatted using Latex to clearly represent equations and operations.

By understanding these concepts, one can approach similar problems involving implicit differentiation and the application of various differentiation rules.