Questions: Find the derivative of the function. y = -12 / (x^(1/3)) dy/dx =

Transcript text: Find the derivative of the function. \[ \begin{array}{l} y=\frac{-12}{\sqrt[3]{x}} \\ \frac{d y}{d x}=\square \end{array} \] $\square$

Solution

Solution Steps

To find the derivative of the function $ y = \frac{-12}{\sqrt[3]{x}} $, we first rewrite the function in a form that is easier to differentiate using power rules. The function can be rewritten as $ y = -12x^{-\frac{1}{3}} $. Then, we apply the power rule for differentiation, which states that if $ y = ax^n $, then $ \frac{dy}{dx} = anx^{n-1} $.

Step 1: Rewrite the Function

The given function is $ y = \frac{-12}{\sqrt[3]{x}} $. We can rewrite this function using exponents as $ y = -12x^{-\frac{1}{3}} $.

Step 2: Apply the Power Rule

To find the derivative, we apply the power rule. The power rule states that if $ y = ax^n $, then $ \frac{dy}{dx} = anx^{n-1} $. For our function, $ a = -12 $ and $ n = -\frac{1}{3} $.

Step 3: Calculate the Derivative

Using the power rule: \[ \frac{dy}{dx} = -12 \times \left(-\frac{1}{3}\right) \times x^{-\frac{1}{3} - 1} = 4x^{-\frac{4}{3}} \]

Step 4: Simplify the Expression

The expression $ 4x^{-\frac{4}{3}} $ can be rewritten as: \[ \frac{4}{x^{\frac{4}{3}}} \]