Differentiate `(1)/( x^3)` with respect to `x` from definition.

Answer

To differentiate \( f(x) = \frac{1}{x^3} \) with respect to \( x \) using the definition of the derivative (first principles), we follow these steps:

### Step 1: Define the function and the increment
Let \( f(x) = \frac{1}{x^3} \). We also define \( f(x + h) \):
\[ f(x + h) = \frac{1}{(x + h)^3} \]

### Step 2: Write the formula for the derivative
The derivative \( f'(x) \) is given by the limit:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]
Substituting the expressions for \( f(x + h) \) and \( f(x) \):
\[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x + h)^3} - \frac{1}{x^3}}{h} \]

### Step 3: Simplify the expression
To simplify the expression, we need a common denominator for the two fractions in the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{x^3 - (x + h)^3}{x^3 (x + h)^3}}{h} \]
This can be rewritten as:
\[ f'(x) = \lim_{h \to 0} \frac{x^3 - (x^3 + 3x^2h + 3xh^2 + h^3)}{h \cdot x^3 (x + h)^3} \]
Simplifying the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{-3x^2h - 3xh^2 - h^3}{h \cdot x^3 (x + h)^3} \]

### Step 4: Factor out \( h \) from the numerator
Factoring \( h \) out of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{-h(3x^2 + 3xh + h^2)}{h \cdot x^3 (x + h)^3} \]
Cancelling \( h \) from the numerator and denominator:
\[ f'(x) = \lim_{h \to 0} \frac{-(3x^2 + 3xh + h^2)}{x^3 (x + h)^3} \]

### Step 5: Apply the limit
Now we can evaluate the limit as \( h \) approaches 0:
\[ f'(x) = \frac{-3x^2}{x^3 (x + 0)^3} = \frac{-3x^2}{x^3 \cdot x^3} = \frac{-3x^2}{x^6} \]
This simplifies to:
\[ f'(x) = -\frac{3}{x^4} \]

### Final Result
Thus, the derivative of \( f(x) = \frac{1}{x^3} \) is:
\[ f'(x) = -\frac{3}{x^4} \]

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