Differentiate from first principles:
10. `(1)/( x^((3)/(2)) )`

Answer

To differentiate the function \( f(x) = \frac{1}{x^{\frac{3}{2}}} \) from first principles, we will follow the definition of the derivative:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 1: Write down the function and the increment
Given:
\[ f(x) = \frac{1}{x^{\frac{3}{2}}} \]
We need to find \( f(x+h) \):
\[ f(x+h) = \frac{1}{(x+h)^{\frac{3}{2}}} \]

### Step 2: Substitute into the derivative formula
Now substitute \( f(x) \) and \( f(x+h) \) into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)^{\frac{3}{2}}} - \frac{1}{x^{\frac{3}{2}}}}{h} \]

### Step 3: Combine the fractions in the numerator
To combine the fractions in the numerator, we need a common denominator:
\[ f'(x) = \lim_{h \to 0} \frac{x^{\frac{3}{2}} - (x+h)^{\frac{3}{2}}}{h \cdot (x+h)^{\frac{3}{2}} \cdot x^{\frac{3}{2}}} \]

### Step 4: Simplify the numerator
Now we simplify the numerator \( x^{\frac{3}{2}} - (x+h)^{\frac{3}{2}} \). Using the binomial expansion for \( (x+h)^{\frac{3}{2}} \):
\[ (x+h)^{\frac{3}{2}} = x^{\frac{3}{2}} + \frac{3}{2} x^{\frac{1}{2}} h + O(h^2) \]
Thus,
\[ x^{\frac{3}{2}} - (x+h)^{\frac{3}{2}} = -\frac{3}{2} x^{\frac{1}{2}} h + O(h^2) \]

### Step 5: Substitute back into the limit
Substituting back into the limit gives:
\[ f'(x) = \lim_{h \to 0} \frac{-\frac{3}{2} x^{\frac{1}{2}} h + O(h^2)}{h \cdot (x+h)^{\frac{3}{2}} \cdot x^{\frac{3}{2}}} \]

### Step 6: Cancel \( h \) in the numerator and denominator
We can cancel \( h \) from the numerator and denominator:
\[ f'(x) = \lim_{h \to 0} \frac{-\frac{3}{2} x^{\frac{1}{2}} + O(h)}{(x+h)^{\frac{3}{2}} \cdot x^{\frac{3}{2}}} \]

### Step 7: Evaluate the limit as \( h \to 0 \)
As \( h \to 0 \), \( (x+h)^{\frac{3}{2}} \) approaches \( x^{\frac{3}{2}} \):
\[ f'(x) = \frac{-\frac{3}{2} x^{\frac{1}{2}}}{x^{\frac{3}{2}} \cdot x^{\frac{3}{2}}} = \frac{-\frac{3}{2} x^{\frac{1}{2}}}{x^{3}} = -\frac{3}{2} \cdot \frac{1}{x^{\frac{5}{2}}} \]

### Final Result
Thus, the derivative is:
\[ f'(x) = -\frac{3}{2} x^{-\frac{5}{2}} \]