Find `dy/dx if y=9x^90`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = 9x^{90} \), we will use the power rule of differentiation. The power rule states that if \( y = ax^n \), then \( \frac{dy}{dx} = n \cdot ax^{n-1} \), where \( a \) is a constant, \( n \) is the exponent, and \( x \) is the variable.

### Step-by-Step Solution:

1. **Identify the function**:
We have \( y = 9x^{90} \).

2. **Apply the power rule**:
According to the power rule, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = n \cdot a \cdot x^{n-1} \]
Here, \( n = 90 \) and \( a = 9 \).

3. **Calculate \( \frac{dy}{dx} \)**:
Substitute \( n \) and \( a \) into the formula:
\[ \frac{dy}{dx} = 90 \cdot 9 \cdot x^{90-1} \]

4. **Simplify the expression**:
Calculate \( 90 \cdot 9 \):
\[ 90 \cdot 9 = 810 \]
Therefore, we have:
\[ \frac{dy}{dx} = 810x^{89} \]

### Final Answer:
\[ \frac{dy}{dx} = 810x^{89} \]