Question Statement

Find $\frac{dy}{dx}$ given that:

$y={\left(x-\frac{1}{x^2}\right)}^5$

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Background and Explanation

This problem requires the chain rule for differentiating composite functions (functions raised to a power). You will also need the power rule for handling both positive and negative exponents.

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Solution

We are given the function:

$y={\left(x-\frac{1}{x^2}\right)}^5$

This is a composite function where the outer function is $u^5$ and the inner function is $u=x-\frac{1}{x^2}$.

Applying the Chain Rule

The chain rule for differentiation states that:

$\frac{d}{dx}[f(x){]}^n=n[f(x){]}^{n-1}\cdot \frac{d}{dx}[f(x)]$

Applying this to our function:

$\frac{dy}{dx}=5{\left(x-\frac{1}{x^2}\right)}^{5-1}\cdot \frac{d}{dx}\left(x-\frac{1}{x^2}\right)$

Simplifying the exponent:

$\frac{dy}{dx}=5{\left(x-\frac{1}{x^2}\right)}^4\cdot \frac{d}{dx}\left(x-\frac{1}{x^2}\right)$

Differentiating the Inner Function

To differentiate the inner function $x-\frac{1}{x^2}$, first rewrite $\frac{1}{x^2}$ using negative exponents:

$\frac{1}{x^2}=x^{-2}$

So we need to find:

$\frac{d}{dx}\left(x-x^{-2}\right)$

Applying the power rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$ to each term:

• The derivative of $x$ (which is $x^1$) is $1\cdot x^0=1$
• The derivative of $-x^{-2}$ is $-(-2)x^{-2-1}=2x^{-3}$

Therefore:

$\frac{d}{dx}\left(x-x^{-2}\right)=1+2x^{-3}$

Final Simplification

Substituting this back into our chain rule expression:

$\frac{dy}{dx}=5{\left(x-\frac{1}{x^2}\right)}^4\cdot \left(1+2x^{-3}\right)$

Rewriting $2x^{-3}$ as a fraction to match the form in the original expression:

$\frac{dy}{dx}=5{\left(x-\frac{1}{x^2}\right)}^4\left(1+\frac{2}{x^3}\right)$

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Key Formulas or Methods Used

• Chain Rule: $\frac{d}{dx}[f(x){]}^n=n[f(x){]}^{n-1}\cdot \frac{d}{dx}[f(x)]$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Negative Exponent Conversion: $\frac{1}{x^n}=x^{-n}$ (used to simplify differentiation of reciprocal terms)

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Summary of Steps

1. Identify the composite structure: Recognize $y=[f(x){]}^5$ where $f(x)=x-\frac{1}{x^2}$
2. Apply the chain rule: Differentiate the outer power function to get $5(x-\frac{1}{x^2}{)}^4$, keeping the inner function unchanged
3. Prepare the inner function: Rewrite $\frac{1}{x^2}$ as $x^{-2}$ to enable power rule application
4. Differentiate the inner function: Calculate $\frac{d}{dx}(x-x^{-2})=1+2x^{-3}$
5. Combine and simplify: Multiply the results from steps 2 and 4, then convert $x^{-3}$ back to fractional form to obtain the final answer: $5(x-\frac{1}{x^2}{)}^4(1+\frac{2}{x^3})$