Differentiate the following functions:
`(7)/( x^( (2)/(3) ) )`

Answer

To differentiate the function \( f(x) = \frac{7}{x^{\frac{2}{3}}} \), we will follow these steps:

### Step 1: Rewrite the function
We can rewrite the function in a form that is easier to differentiate. The expression \( \frac{7}{x^{\frac{2}{3}}} \) can be rewritten using negative exponents:
\[ f(x) = 7 \cdot x^{-\frac{2}{3}} \]

### Step 2: Apply the constant multiple rule
Since 7 is a constant, we can take it out of the differentiation:
\[ \frac{d}{dx}[f(x)] = 7 \cdot \frac{d}{dx}[x^{-\frac{2}{3}}] \]

### Step 3: Differentiate using the power rule
Now, we will apply the power rule of differentiation, which states that \( \frac{d}{dx}[x^m] = m \cdot x^{m-1} \):
\[ \frac{d}{dx}[x^{-\frac{2}{3}}] = -\frac{2}{3} \cdot x^{-\frac{2}{3} - 1} = -\frac{2}{3} \cdot x^{-\frac{5}{3}} \]

### Step 4: Combine the results
Now, substituting back into our earlier expression:
\[ \frac{d}{dx}[f(x)] = 7 \cdot \left(-\frac{2}{3} \cdot x^{-\frac{5}{3}}\right) \]
\[ = -\frac{14}{3} \cdot x^{-\frac{5}{3}} \]

### Step 5: Rewrite in a more standard form
We can rewrite the expression back into the form of a fraction:
\[ \frac{d}{dx}[f(x)] = -\frac{14}{3} \cdot \frac{1}{x^{\frac{5}{3}}} \]

Thus, the derivative of the function \( f(x) = \frac{7}{x^{\frac{2}{3}}} \) is:
\[ \frac{d}{dx}[f(x)] = -\frac{14}{3} \cdot \frac{1}{x^{\frac{5}{3}}} \]