What is the value of

\[\dfrac{d}{dx}\left(\dfrac{1}{x^4}-\dfrac{3}{x^2}\right)\] at
\[x=1\]?

We are tasked with finding the derivative of the expression

\[
\dfrac{d}{dx}\left(\dfrac{1}{x^4}-\dfrac{3}{x^2}\right)
\]

and then evaluating it at \(x = 1\).

### Step 1: Differentiate the expression
The given expression is:

\[
f(x) = \dfrac{1}{x^4} - \dfrac{3}{x^2}
\]

We will differentiate each term separately.

#### For the first term, \(\dfrac{1}{x^4}\):
\[
\dfrac{d}{dx}\left(\dfrac{1}{x^4}\right) = \dfrac{d}{dx}(x^{-4}) = -4x^{-5}
\]

#### For the second term, \(\dfrac{3}{x^2}\):
\[
\dfrac{d}{dx}\left(\dfrac{3}{x^2}\right) = 3 \cdot \dfrac{d}{dx}(x^{-2}) = 3 \cdot (-2x^{-3}) = -6x^{-3}
\]

### Step 2: Combine the results
Now, we combine the derivatives:

\[
f'(x) = -4x^{-5} - 6x^{-3}
\]

### Step 3: Evaluate at \(x = 1\)
Substitute \(x = 1\) into the derivative:

\[
f'(1) = -4(1)^{-5} - 6(1)^{-3} = -4 - 6 = -10
\]

Thus, the value of the derivative at \(x = 1\) is \(-10\).

### Final Answer:
\[
f'(1) = -10
\]

Would you like any additional details or clarifications on the steps?

Here are 5 related questions to consider:
1. How would the result change if the expression had different powers of \(x\)?
2. What is the second derivative of this function?
3. How does the chain rule apply when differentiating more complex functions?
4. What would the derivative be if the expression included a higher power of \(x\)?
5. How does the power rule for differentiation simplify the process?

**Tip:** Always rewrite expressions with exponents in a simpler form (like negative powers) before differentiating.

What is the value of \( \dfrac{d}{dx} \left( \dfrac{1}{x^4} - \dfrac{3}{x^2} \right) \) at \(x=1\)?

Find the derivative of the function \( \dfrac{1}{x^4} - \dfrac{3}{x^2} \) and evaluate it at \(x = 1\). This step-by-step solution covers differentiation using the power rule, explaining each term and simplifying the result. Suitable for students in Grades 11-12.

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