Questions: u=-1/x^4 => du=? 4 x^-5 dx -4 x^5 dx 4 x^-3 dx -4 x^-3 dx

Transcript text: \[ u=-\frac{1}{x^{4}} \Rightarrow d u=? \] $4 x^{-5} d x$ $-4 x^{5} d x$ $4 x^{-3} d x$ $-4 x^{-3} d x$

Solution

Step 1: Differentiate the Function

Given the function $ u = -\frac{1}{x^4} $, we need to find the differential $ du $.

First, rewrite the function in a more convenient form for differentiation:

\[ u = -x^{-4} \]

Step 2: Apply the Power Rule

Differentiate $ u = -x^{-4} $ with respect to $ x $ using the power rule, which states that if $ u = x^n $, then $ \frac{du}{dx} = n \cdot x^{n-1} $.

\[ \frac{du}{dx} = -(-4)x^{-4-1} = 4x^{-5} \]

Step 3: Express the Differential

The differential $ du $ is given by:

\[ du = \frac{du}{dx} \cdot dx = 4x^{-5} \cdot dx \]

The correct differential is:

\[ \boxed{4x^{-5} \, dx} \]

Thus, the answer is the first option: $ 4x^{-5} \, dx $.

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