To find the derivative of the function $y = 8\sqrt{x} + 7x^{\frac{4}{5}}$ with respect to $x$, we can proceed as follows:

1. Rewrite the terms for easier differentiation: $y = 8x^{\frac{1}{2}} + 7x^{\frac{4}{5}}$

2. Differentiate each term separately using the power rule, $\frac{d}{dx}(x^n) = n x^{n-1}$.

Step-by-Step Solution:

• Differentiate the first term $8x^{\frac{1}{2}}$: $\frac{d}{dx}(8x^{\frac{1}{2}}) = 8 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = 4x^{-\frac{1}{2}} = \frac{4}{\sqrt{x}}$

• Differentiate the second term $7x^{\frac{4}{5}}$: $\frac{d}{dx}(7x^{\frac{4}{5}}) = 7 \cdot \frac{4}{5} x^{\frac{4}{5} - 1} = \frac{28}{5} x^{-\frac{1}{5}}$

3. Combine the results to write the derivative: $\frac{dy}{dx} = \frac{4}{\sqrt{x}} + \frac{28}{5} x^{-\frac{1}{5}}$

So, the final answer is: $\frac{dy}{dx} = \frac{4}{\sqrt{x}} + \frac{28}{5} x^{-\frac{1}{5}}$

Would you like a detailed explanation of any step or have any questions?

---

Related Questions

1. How would the derivative change if the powers were different, say $x^{3/5}$ instead of $x^{4/5}$?
2. How would we differentiate a function like $y = 8x^{1/3} + 7x^{2/3}$?
3. What is the second derivative of $y = 8\sqrt{x} + 7x^{4/5}$?
4. How do we find the derivative of functions with roots and fractional exponents in general?
5. What if we had a negative exponent, like $y = 8x^{-1/2} + 7x^{-4/5}$?

Tip

Always simplify roots and fractional powers before differentiating, as it makes applying rules easier.