Questions: d/dx (1/(3x-1)^5) =

d/dx (1/(3x-1)^5) =
Transcript text: \[ \frac{d}{d x} \frac{1}{(3 x-1)^{5}}= \]
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Solution

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Solution Steps

Step 1: Identify the function and the rule to apply

The given function is \( \frac{d}{dx} \left( (3x - 1)^5 \right) \). This is a composite function, so we will use the chain rule to differentiate it.

Step 2: Apply the chain rule

The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \). Here, let \( u = 3x - 1 \) and \( f(u) = u^5 \).

Step 3: Differentiate the outer function

Differentiate \( f(u) = u^5 \) with respect to \( u \): \[ f'(u) = 5u^4 \]

Step 4: Differentiate the inner function

Differentiate \( u = 3x - 1 \) with respect to \( x \): \[ \frac{du}{dx} = 3 \]

Step 5: Combine the results

Using the chain rule, combine the derivatives: \[ \frac{d}{dx} \left( (3x - 1)^5 \right) = 5(3x - 1)^4 \cdot 3 \]

Final Answer

\[ \frac{d}{dx} \left( (3x - 1)^5 \right) = 15(3x - 1)^4 \]

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