Question Statement

Given the function:

$y=\frac{2}{x^4}$

Find the first derivative $\frac{dy}{dx}$ and the second derivative $\frac{d^{2}y}{d{x}^2}$.

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Background and Explanation

This problem requires differentiating a rational function by first converting it to negative exponent form. You will apply the power rule for differentiation successively to find higher-order derivatives.

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Solution

Begin by expressing the function using negative exponents, which makes applying the power rule straightforward:

$y=2x^{-4}$

Finding the First Derivative

Differentiate with respect to $x$ using the power rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$:

$\begin{array}{rl}\frac{dy}{dx}& =2\frac{d}{dx}\left(x^{-4}\right)\\ & =2\left(-4x^{-4-1}\right)\\ \frac{dy}{dx}& =-8x^{-5}\end{array}$

Finding the Second Derivative

Differentiate the first derivative again with respect to $x$:

$\begin{array}{rl}\frac{d^{2}y}{d{x}^2}& =-8\frac{d}{dx}\left(x^{-5}\right)\\ & =-8\left(-5x^{-5-1}\right)\\ & =40x^{-6}\\ & =\frac{40}{x^6}\end{array}$

Thus, the first derivative is $\frac{dy}{dx}=-8x^{-5}$ (or $-\frac{8}{x^5}$) and the second derivative is $\frac{d^{2}y}{d{x}^2}=\frac{40}{x^6}$.

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Key Formulas or Methods Used

• Negative Exponent Conversion: $\frac{a}{x^n}=a{x}^{-n}$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Constant Multiple Rule: $\frac{d}{dx}[c\cdot f(x)]=c\cdot \frac{d}{dx}[f(x)]$

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Summary of Steps

1. Rewrite the function as $y=2x^{-4}$ using negative exponents
2. Apply the power rule to find the first derivative: $\frac{dy}{dx}=2(-4)x^{-5}=-8x^{-5}$
3. Differentiate the result again using the power rule: $\frac{d^{2}y}{d{x}^2}=-8(-5)x^{-6}=40x^{-6}$
4. Convert the final answer back to positive exponent form: $\frac{40}{x^6}$