Question Statement

Find the second derivative of the function $y=20x^{-3}$ with respect to $x$.

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Background and Explanation

This problem requires applying the power rule for differentiation twice to find the second derivative $\frac{d^{2}y}{d{x}^2}$. You should be familiar with differentiating power functions of the form $x^n$ and handling negative exponents.

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Solution

Finding the First Derivative

Given the function: $y=20x^{-3}$

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

$\begin{array}{rl}\frac{dy}{dx}& =20\cdot \frac{d}{dx}(x^{-3})\\ & =20(-3x^{-3-1})\\ & =20(-3x^{-4})\\ \frac{dy}{dx}& =-60x^{-4}\end{array}$

Finding the Second Derivative

Now differentiate $\frac{dy}{dx}=-60x^{-4}$ with respect to $x$ again to find the second derivative:

$\begin{array}{rl}\frac{d^{2}y}{d{x}^2}& =-60\cdot \frac{d}{dx}(x^{-4})\\ & =-60(-4x^{-4-1})\\ & =-60(-4x^{-5})\\ & =240x^{-5}\end{array}$

This can also be written with a positive exponent in the denominator: $\frac{d^{2}y}{d{x}^2}=\frac{240}{x^5}$

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Key Formulas or Methods Used

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

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Summary of Steps

1. Identify the function: $y=20x^{-3}$
2. Apply power rule to find the first derivative: multiply by the exponent $(-3)$ and reduce the exponent by $1$ to get $\frac{dy}{dx}=-60x^{-4}$
3. Differentiate again using the power rule on $-60x^{-4}$: multiply by $(-4)$ and reduce the exponent by $1$
4. Simplify to obtain $\frac{d^{2}y}{d{x}^2}=240x^{-5}=\frac{240}{x^5}$