Find $\frac{d y}{d x}$, if $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$

Step 1: Differentiate implicitly

To find $\frac{dy}{dx}$, we need to differentiate both sides of the given equation with respect to $x$. Since $y$ is a function of $x$, we will use the chain rule for the term involving $y$. The term $a^{\frac{2}{3}}$ is a constant, so its derivative will be zero.

Step 2: Apply power rule and chain rule

Applying the power rule, $\frac{d}{dx}(x^n) = nx^{n-1}$, to $x^{\frac{2}{3}}$ gives $\frac{2}{3}x^{-\frac{1}{3}}$. For $y^{\frac{2}{3}}$, we use the chain rule, which results in $\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}$. The derivative of the constant $a^{\frac{2}{3}}$ is $0$.

Step 3: Simplify the exponents

We simplify the exponents: $\frac{2}{3} - 1 = -\frac{1}{3}$. This gives us the simplified form of the differentiated equation.

Step 4: Isolate $\frac{dy}{dx}$ term

To solve for $\frac{dy}{dx}$, we first move the term not containing $\frac{dy}{dx}$ to the right side of the equation.

Step 5: Solve for $\frac{dy}{dx}$

Divide both sides by $\frac{2}{3}y^{-\frac{1}{3}}$ to solve for $\frac{dy}{dx}$. We can then simplify the expression using the property $a^{-n} = \frac{1}{a^n}$ and $a^{\frac{1}{n}} = \sqrt[n]{a}$.

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