Find the derivatives of the following (9-10) at any point of their domains :
`f(x)=x^(-4)(3-4x^(-5))`

Answer

To find the derivative of the function \( f(x) = x^{-4}(3 - 4x^{-5}) \), we will follow these steps:

### Step 1: Expand the Function
First, we will distribute \( x^{-4} \) to both terms inside the parentheses.

\[ f(x) = x^{-4} \cdot 3 - x^{-4} \cdot 4x^{-5} \]

This simplifies to:

\[ f(x) = 3x^{-4} - 4x^{-4} \cdot x^{-5} \]

### Step 2: Simplify the Expression
Next, we can simplify the second term using the property of exponents \( a^m \cdot a^n = a^{m+n} \):

\[ f(x) = 3x^{-4} - 4x^{-4-5} = 3x^{-4} - 4x^{-9} \]

### Step 3: Differentiate the Function
Now we will differentiate \( f(x) \) with respect to \( x \). We will use the power rule for differentiation, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \).

Differentiating each term:

1. For \( 3x^{-4} \):
\[ \frac{d}{dx}(3x^{-4}) = 3 \cdot (-4)x^{-4-1} = -12x^{-5} \]

2. For \( -4x^{-9} \):
\[ \frac{d}{dx}(-4x^{-9}) = -4 \cdot (-9)x^{-9-1} = 36x^{-10} \]

### Step 4: Combine the Derivatives
Now we can combine the results from the differentiation:

\[ f'(x) = -12x^{-5} + 36x^{-10} \]

### Step 5: Rewrite the Derivative
Finally, we can rewrite the derivative in a more standard form:

\[ f'(x) = -\frac{12}{x^5} + \frac{36}{x^{10}} \]

### Final Answer
Thus, the derivative of the function \( f(x) = x^{-4}(3 - 4x^{-5}) \) is:

\[ f'(x) = -\frac{12}{x^5} + \frac{36}{x^{10}} \]

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