7. Find the derivative for the following functions a. $$ y=\frac{1}{4} x^{4}+9 x-6 \sqrt{x^{3}} $$

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To find the derivative of the function:

$$ y = \frac{1}{4} x^{4} + 9x - 6 \sqrt{x^{3}} $$

Let's differentiate each term separately.

1. **First Term: $$\frac{1}{4}x^4$$**
   
   $$
   \frac{d}{dx} \left( \frac{1}{4} x^{4} \right) = \frac{1}{4} \cdot 4x^{3} = x^{3}
   $$

2. **Second Term: $$9x$$**
   
   $$
   \frac{d}{dx} (9x) = 9
   $$

3. **Third Term: $$-6\sqrt{x^{3}}$$**

First, rewrite $$\sqrt{x^{3}}$$ as $$x^{3/2}$$:
   
   $$
   -6 \sqrt{x^{3}} = -6x^{3/2}
   $$
   
   Now, differentiate:
   
   $$
   \frac{d}{dx} \left( -6x^{3/2} \right) = -6 \cdot \frac{3}{2} x^{1/2} = -9x^{1/2} = -9\sqrt{x}
   $$

**Combine all the derivatives:**

$$
\frac{dy}{dx} = x^{3} + 9 - 9\sqrt{x}
$$

**Final Answer:**

After simplifying, the derivative is x³ + 9 minus nine times √x. Thus,

y′ = x³ + 9 − 9 √x
The derivative of the function $$ y = \frac{1}{4} x^{4} + 9x - 6 \sqrt{x^{3}} $$ is: $$ y' = x^{3} + 9 - 9\sqrt{x} $$

7. Find the derivative for the following functions a. \( y=\frac{1}{4} x^{4}+9 x-6 \sqrt{x^{3}} \)

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