DIfferentiate (i) `3x^(-5) ` (ii) `1/(5x)` (iii) ` root3(x^(2))`

Answer

Let's differentiate the given functions step by step.

### Part (i): Differentiate \(3x^{-5}\)

1. **Identify the function**: The function is \(3x^{-5}\).
2. **Apply the power rule**: The power rule states that \(\frac{d}{dx} x^n = n x^{n-1}\).
3. **Differentiate**:
\[ \frac{d}{dx}(3x^{-5}) = 3 \cdot \frac{d}{dx}(x^{-5}) = 3 \cdot (-5)x^{-5-1} = -15x^{-6} \]

### Solution for Part (i):
\[ \frac{d}{dx}(3x^{-5}) = -15x^{-6} \]

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### Part (ii): Differentiate \(\frac{1}{5x}\)

1. **Rewrite the function**: \(\frac{1}{5x}\) can be rewritten as \(\frac{1}{5} x^{-1}\).
2. **Apply the power rule**:
\[ \frac{d}{dx}\left(\frac{1}{5} x^{-1}\right) = \frac{1}{5} \cdot \frac{d}{dx}(x^{-1}) = \frac{1}{5} \cdot (-1)x^{-1-1} = -\frac{1}{5}x^{-2} \]

### Solution for Part (ii):
\[ \frac{d}{dx}\left(\frac{1}{5x}\right) = -\frac{1}{5}x^{-2} \]

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### Part (iii): Differentiate \(\sqrt[3]{x^2}\)

1. **Rewrite the function**: The cube root can be expressed as a power: \(\sqrt[3]{x^2} = x^{\frac{2}{3}}\).
2. **Apply the power rule**:
\[ \frac{d}{dx}(x^{\frac{2}{3}}) = \frac{2}{3} x^{\frac{2}{3}-1} = \frac{2}{3} x^{-\frac{1}{3}} \]

### Solution for Part (iii):
\[ \frac{d}{dx}\left(\sqrt[3]{x^2}\right) = \frac{2}{3}x^{-\frac{1}{3}} \]

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### Summary of Solutions:
1. \(\frac{d}{dx}(3x^{-5}) = -15x^{-6}\)
2. \(\frac{d}{dx}\left(\frac{1}{5x}\right) = -\frac{1}{5}x^{-2}\)
3. \(\frac{d}{dx}\left(\sqrt[3]{x^2}\right) = \frac{2}{3}x^{-\frac{1}{3}}\)

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