Find the derivative of f(x) = 3x^(5/2) + x^(-4) - 3x^(1/2)

Steps to Solve

1. Apply the power rule to the first term

The power rule states that if $f(x) = ax^n$, then $f'(x) = nax^{n-1}$. For the first term, $3x^{5/2}$, we have $a = 3$ and $n = 5/2$. Applying the power rule: $$(3x^{5/2})' = \frac{5}{2} \cdot 3x^{\frac{5}{2}-1} = \frac{15}{2}x^{\frac{3}{2}}$$

2. Apply the power rule to the second term

For the second term, $x^{-4}$, we have $a = 1$ and $n = -4$. Applying the power rule: $$(x^{-4})' = -4 \cdot x^{-4-1} = -4x^{-5}$$

3. Apply the power rule to the third term

For the third term, $-3x^{1/2}$, we have $a = -3$ and $n = 1/2$. Applying the power rule: $$(-3x^{1/2})' = \frac{1}{2} \cdot (-3)x^{\frac{1}{2}-1} = -\frac{3}{2}x^{-\frac{1}{2}}$$

4. Combine the derivatives of each term

Now, we sum up the derivatives of each term to find the derivative of the entire function: $$f'(x) = \frac{15}{2}x^{\frac{3}{2}} - 4x^{-5} - \frac{3}{2}x^{-\frac{1}{2}}$$