Given the function y, equals, start fraction, 1, divided by, 6, x, squared, end fraction, commay= 
6x 
2
 
1
​
 , find start fraction, d, y, divided by, d, x, end fraction, . 
dx
dy
​
 . Express your answer in simplest form without using negative exponents.

To find \(\frac{dy}{dx}\), the derivative of the function \(y = \frac{1}{6x^2}\), we will apply the power rule of differentiation. First, rewrite the function to a more convenient form:

\[
y = \frac{1}{6}x^{-2}
\]

Now, differentiate using the power rule \(\frac{d}{dx}x^n = nx^{n-1}\):

\[
\frac{dy}{dx} = \frac{1}{6} \cdot (-2) x^{-3}
\]

Simplify:

\[
\frac{dy}{dx} = -\frac{2}{6} x^{-3}
\]

\[
\frac{dy}{dx} = -\frac{1}{3} x^{-3}
\]

Now, express the answer without negative exponents:

\[
\frac{dy}{dx} = -\frac{1}{3x^3}
\]

Thus, the derivative is:

\[
\frac{dy}{dx} = -\frac{1}{3x^3}
\]

Would you like more details or have any questions? Here are 5 related questions:

1. How do you apply the power rule in differentiation?
2. Can you differentiate \(y = x^{-4}\)?
3. What is the significance of negative exponents in derivatives?
4. How does one differentiate functions with constants like \(\frac{1}{6}\)?
5. How would you handle the chain rule if the function was more complex?

**Tip:** Always simplify the function into a form that is easier to differentiate, such as converting fractions to negative exponents before applying rules.

Given the function y = 1/(6x^2), find dy/dx. Express your answer in simplest form without using negative exponents.

Learn how to find the derivative of the function y = 1/(6x^2) using the power rule of differentiation. This detailed solution explains key concepts, including handling negative exponents and simplifying the final expression. Ideal for calculus students in Grades 11-12 or introductory college-level courses.

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