The image shows the function:

$y = \sqrt{x} + \sqrt[3]{x}$

To calculate the derivative of this function with respect to $x$, let's proceed step by step:

1. Rewrite the square root and cube root terms in exponent form: $y = x^{1/2} + x^{1/3}$

2. Now apply the power rule of differentiation, which states that for $f(x) = x^n$, the derivative is $f'(x) = nx^{n-1}$.

   • Derivative of $x^{1/2}$: $\frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$

   • Derivative of $x^{1/3}$: $\frac{d}{dx} \left( x^{1/3} \right) = \frac{1}{3}x^{-\frac{2}{3}} = \frac{1}{3x^{2/3}}$

3. Therefore, the derivative of the function is: $\frac{dy}{dx} = \frac{1}{2\sqrt{x}} + \frac{1}{3x^{2/3}}$

Would you like more details or have any questions?

Here are five related questions:

1. How do you apply the chain rule to more complex functions involving square and cube roots?
2. What happens if the function involves higher roots or fractional exponents?
3. How do you differentiate a sum of more than two functions with roots?
4. Can you find the second derivative of this function?
5. How do these derivative rules apply to real-world applications like physics or economics?

Tip: When differentiating, it's often helpful to rewrite roots or radicals as fractional exponents to simplify the application of rules like the power rule.