Find the derivative of the function f(x) = 3√x - x^(-2). Also, find the equation of the normal to the graph of f(x) = x^3 - 5x.

Steps to Solve

1. Rewrite the function using exponents

To easily apply the power rule, rewrite the square root and the negative exponent as follows:

$f(x) = 3x^{\frac{1}{2}} - x^{-2}$

2. Apply the power rule to the first term

The power rule states that if $f(x) = ax^n$, then $f'(x) = n \cdot ax^{n-1}$. Applying this to the first term:

$\frac{d}{dx}(3x^{\frac{1}{2}}) = 3 \cdot \frac{1}{2} x^{\frac{1}{2}-1} = \frac{3}{2}x^{-\frac{1}{2}}$

3. Apply the power rule to the second term

Applying the power rule to the second term:

$\frac{d}{dx}(-x^{-2}) = -(-2)x^{-2-1} = 2x^{-3}$

4. Combine the derivatives

Using the difference rule, combine the derivatives of the two terms:

$f'(x) = \frac{3}{2}x^{-\frac{1}{2}} + 2x^{-3}$

5. Rewrite with radicals and positive exponents

Rewrite the derivative using radicals and positive exponents to match the answer in the image:

$f'(x) = \frac{3}{2\sqrt{x}} + \frac{2}{x^3}$ or $f'(x) = \frac{3}{2\sqrt{x}} + 2x^{-3}$ This is the exact solution in the image