Question Statement

Find the fifth derivative of the function:

$y=\frac{2}{x}$

Determine $\frac{d^{5}y}{d{x}^5}$.

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Background and Explanation

This problem requires repeated application of the power rule for differentiation. First, rewrite the rational function using negative exponents to enable direct application of the rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$.

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Solution

Begin by rewriting the function in power form to facilitate differentiation:

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

First Derivative:

Differentiate with respect to $x$ using the power rule:

$\begin{array}{rl}\frac{dy}{dx}& =2\frac{d}{dx}\left(x^{-1}\right)\\ & =2\left(-1\cdot x^{-1-1}\right)\\ \frac{dy}{dx}& =-2x^{-2}\end{array}$

Second Derivative:

Differentiate again with respect to $x$:

$\begin{array}{rl}\frac{d^{2}y}{d{x}^2}& =-2\left(-2x^{-3}\right)\\ \frac{d^{2}y}{d{x}^2}& =4x^{-3}\end{array}$

Third Derivative:

Differentiate again with respect to $x$:

$\begin{array}{rl}\frac{d^{3}y}{d{x}^3}& =4\left(-3x^{-4}\right)\\ \frac{d^{3}y}{d{x}^3}& =-12x^{-4}\end{array}$

Fourth Derivative:

Differentiate again with respect to $x$:

$\begin{array}{rl}\frac{d^{4}y}{d{x}^4}& =-12\left(-4x^{-5}\right)\\ \frac{d^{4}y}{d{x}^4}& =48x^{-5}\end{array}$

Fifth Derivative:

Differentiate once more with respect to $x$:

$\begin{array}{rl}\frac{d^{5}y}{d{x}^5}& =48\left(-5x^{-6}\right)\\ & =-240x^{-6}\\ \frac{d^{5}y}{d{x}^5}& =\frac{-240}{x^6}\end{array}$

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Key Formulas or Methods Used

• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Negative Exponent Conversion: $\frac{1}{x^n}=x^{-n}$
• Constant Multiple Rule: $\frac{d}{dx}[cf(x)]=c\frac{d}{dx}f(x)$
• Pattern Recognition: Each differentiation multiplies the coefficient by the current negative exponent and reduces the exponent by 1 (e.g., $-2\times -2=4$, $4\times -3=-12$, etc.)

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Summary of Steps

1. Rewrite the function: $y=2x^{-1}$
2. Apply power rule for first derivative: $\frac{dy}{dx}=-2x^{-2}$
3. Second derivative: $\frac{d^{2}y}{d{x}^2}=4x^{-3}$ (multiply $-2$ by $-2$)
4. Third derivative: $\frac{d^{3}y}{d{x}^3}=-12x^{-4}$ (multiply $4$ by $-3$)
5. Fourth derivative: $\frac{d^{4}y}{d{x}^4}=48x^{-5}$ (multiply $-12$ by $-4$)
6. Fifth derivative: $\frac{d^{5}y}{d{x}^5}=-240x^{-6}=-\frac{240}{x^6}$ (multiply $48$ by $-5$)