Questions: Order Derivatives Find (d^2 y / d x^2). y=1/x^3 d^2 y / dx^2 = □

Transcript text: -Order Derivatives Find $\frac{d^{2} y}{d x^{2}}$. \[ \begin{array}{r} y=\frac{1}{x^{3}} \\ \frac{d^{2} y}{d x^{2}}=\square \end{array} \]

Solution

Solution Steps

To find the second derivative of the function $ y = \frac{1}{x^3} $, we first need to find the first derivative using the power rule. Once we have the first derivative, we apply the power rule again to find the second derivative.

Step 1: Find the First Derivative

To find the first derivative of $ y = \frac{1}{x^3} $, we apply the power rule. The power rule states that if $ y = x^n $, then $ \frac{dy}{dx} = nx^{n-1} $.

For $ y = x^{-3} $, the first derivative is: \[ \frac{dy}{dx} = -3x^{-4} = -\frac{3}{x^4} \]

Step 2: Find the Second Derivative

Next, we find the second derivative by differentiating the first derivative. Again, we use the power rule.

For $ \frac{dy}{dx} = -3x^{-4} $, the second derivative is: \[ \frac{d^2y}{dx^2} = 12x^{-5} = \frac{12}{x^5} \]

Final Answer

The second derivative of $ y = \frac{1}{x^3} $ is: \[ \boxed{\frac{d^2y}{dx^2} = \frac{12}{x^5}} \]

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