Questions: Find dy/dx. y=x^(-6) dy/dx=-(6/x^7)

Solution Steps

To find the derivative of the function \( y = x^{-6} \), we will use the power rule for differentiation. The power rule states that if \( y = x^n \), then \( \frac{d y}{d x} = n \cdot x^{n-1} \).

1. Identify the exponent \( n \) in the function \( y = x^{-6} \).
2. Apply the power rule: \( \frac{d y}{d x} = n \cdot x^{n-1} \).
3. Simplify the expression to get the final derivative.

Given the function: \[ y = x^{-6} \]

Using the power rule for differentiation, which states: \[ \frac{d}{dx} x^n = n \cdot x^{n-1} \] we identify \( n = -6 \).

Applying the power rule: \[ \frac{d y}{d x} = -6 \cdot x^{-6-1} \] \[ \frac{d y}{d x} = -6 \cdot x^{-7} \]

Rewrite the expression using positive exponents: \[ \frac{d y}{d x} = -\frac{6}{x^7} \]

Final Answer