(b) Use appropriate rules of differentiation to differentiate the following functions. (i) $$ f(x)=\left(x^{5}-3 x^{-2}+3\right)\left(x^{\frac{5}{2}}-4 x\right) $$ [TUF

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To differentiate the function 
$$ f(x) = \left(x^{5} - 3x^{-2} + 3\right)\left(x^{\frac{5}{2}} - 4x\right), $$
we can use the **product rule** of differentiation. The product rule states that if you have two functions $$ u(x) $$ and $$ v(x) $$, then the derivative of their product is:
$$ (uv)' = u'v + uv'. $$

Let's apply this step-by-step:

1. **Identify the two functions:**
   $$ u(x) = x^{5} - 3x^{-2} + 3 $$
   $$ v(x) = x^{\frac{5}{2}} - 4x $$

2. **Find the derivatives of each function:**
   
   - **Derivative of $$ u(x) $$:**
     $$
     u'(x) = \frac{d}{dx}\left(x^{5}\right) - \frac{d}{dx}\left(3x^{-2}\right) + \frac{d}{dx}(3) = 5x^{4} + 6x^{-3}
     $$
   
   - **Derivative of $$ v(x) $$:**
     $$
     v'(x) = \frac{d}{dx}\left(x^{\frac{5}{2}}\right) - \frac{d}{dx}(4x) = \frac{5}{2}x^{\frac{3}{2}} - 4
     $$

3. **Apply the product rule:**
   $$
   f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
   $$
   Substituting the expressions we found:
   $$
   f'(x) = \left(5x^{4} + 6x^{-3}\right)\left(x^{\frac{5}{2}} - 4x\right) + \left(x^{5} - 3x^{-2} + 3\right)\left(\frac{5}{2}x^{\frac{3}{2}} - 4\right)
   $$

4. **Simplify (optional):**
   
   While the above form is correct, you can also expand it further if needed:
   $$
   \begin{align*}
   f'(x) &= \left(5x^{4} \cdot x^{\frac{5}{2}}\right) + \left(5x^{4} \cdot (-4x)\right) + \left(6x^{-3} \cdot x^{\frac{5}{2}}\right) + \left(6x^{-3} \cdot (-4x)\right) \
   &\quad + \left(x^{5} \cdot \frac{5}{2}x^{\frac{3}{2}}\right) + \left(x^{5} \cdot (-4)\right) + \left(-3x^{-2} \cdot \frac{5}{2}x^{\frac{3}{2}}\right) + \left(-3x^{-2} \cdot (-4)\right) + \left(3 \cdot \frac{5}{2}x^{\frac{3}{2}}\right) + \left(3 \cdot (-4)\right) \
   &= \frac{15}{2}x^{\frac{13}{2}} - 20x^{5} + 6x^{\frac{1}{2}} - 24x^{-2} + \frac{5}{2}x^{\frac{13}{2}} - 4x^{5} - \frac{15}{2}x^{\frac{-1}{2}} + 12x^{-2} + \frac{15}{2}x^{\frac{3}{2}} - 12 \
   &= \left(\frac{15}{2} + \frac{5}{2}\right)x^{\frac{13}{2}} + (-20 - 4)x^{5} + 6x^{\frac{1}{2}} + \frac{15}{2}x^{\frac{3}{2}} - \frac{15}{2}x^{-\frac{1}{2}} - 12
   \end{align*}
   $$
   Which simplifies to:
   $$
   f'(x) = 10x^{\frac{13}{2}} - 24x^{5} + 6x^{\frac{1}{2}} + \frac{15}{2}x^{\frac{3}{2}} - \frac{15}{2}x^{-\frac{1}{2}} - 12.
   $$
   
   However, in many cases, leaving the derivative in the factored product rule form is acceptable and often preferred for simplicity.

**Final Answer:**

Using the product rule, the derivative is

f ′(x) = (5x⁴ + 6x^–3)(x^(5⁄2) – 4x) + (x⁵ – 3x^–2 + 3)( (5/2)x^(3⁄2) – 4 ).

In symbols,

f′(x) = (5x^4 + 6x^{-3})(x^{5/2} - 4x) + (x^5 - 3x^{-2} + 3)\left(\frac{5}{2}x^{3/2} - 4\right).
The derivative of the function $$ f(x) = \left(x^{5} - 3x^{-2} + 3\right)\left(x^{\frac{5}{2}} - 4x\right) $$ is: $$ f'(x) = (5x^{4} + 6x^{-3})(x^{\frac{5}{2}} - 4x) + (x^{5} - 3x^{-2} + 3)\left(\frac{5}{2}x^{\frac{3}{2}} - 4\right) $$

(b) Use appropriate rules of differentiation to differentiate the following functions. (i) \( f(x)=\left(x^{5}-3 x^{-2}+3\right)\left(x^{\frac{5}{2}}-4 x\right) \) [TUF

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