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

Question

UpStudy AI · Accepted Answer

To find the derivative of the function:

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

**Step 1: Simplify the Function**

First, simplify the constant terms:

$$
y = \frac{1}{4} x^{4} + 9x - 7
$$

**Step 2: Differentiate Term by Term**

We'll apply the power rule for differentiation, which states that:

$$
\frac{d}{dx} \left( x^n \right) = n x^{n-1}
$$

- **First Term:** $$\frac{1}{4} x^{4}$$

$$
  \frac{d}{dx} \left( \frac{1}{4} x^{4} \right) = \frac{1}{4} \times 4x^{3} = x^{3}
  $$

- **Second Term:** $$9x$$

$$
  \frac{d}{dx} (9x) = 9
  $$

- **Third Term:** $$-7$$ (a constant)

$$
  \frac{d}{dx} (-7) = 0
  $$

**Step 3: Combine the Derivatives**

Adding up the derivatives of each term:

$$
\frac{dy}{dx} = x^{3} + 9 + 0 = x^{3} + 9
$$

**Final Answer:**

$$
\boxed{ \frac{dy}{dx} = x^{3} + 9 }
$$
The derivative of $$ y = \frac{1}{4} x^{4} + 9x - 7 $$ is $$ \frac{dy}{dx} = x^{3} + 9 $$.

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