# Learning Derivative Rules for Polynomials

Polynomials are a fundamental class of functions in calculus, and understanding how to differentiate them is a key skill. Here, we'll explore the rules for finding the derivatives of polynomial functions.

## Basic Concept

A polynomial function is generally of the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where each $a_i$ is a constant, and $n$ is a non-negative integer.

### Power Rule

The most basic rule for differentiating polynomials is the Power Rule. If you have a term $ax^n$, its derivative is given by:

$$
\frac{d}{dx}(ax^n) = n \cdot ax^{n-1}
$$

### Applying the Power Rule

#### Example 1: Differentiate $y = 5x^3$

- **Solution**:
$$
\frac{dy}{dx} = 3 \cdot 5x^{3-1} = 15x^2
$$

#### Example 2: Differentiate $P(x) = 4x^4 - 2x^2 + 7$

- **Solution**:
$$
P'(x) = \frac{d}{dx}(4x^4) - \frac{d}{dx}(2x^2) + \frac{d}{dx}(7)
$$
$$
= 4 \cdot 4x^{4-1} - 2 \cdot 2x^{2-1} + 0
$$
$$
= 16x^3 - 4x
$$

### Sum Rule

When differentiating a polynomial, you differentiate each term separately. This is known as the Sum Rule.

#### Example: Differentiate $f(x) = x^5 - 3x^3 + x - 9$

- **Solution**:
$$
f'(x) = 5x^4 - 9x^2 + 1
$$

### Constant Rule

A constant term (like $5$, $-3$, or $a_0$ in a polynomial) differentiates to zero.

#### Example: Differentiate $g(x) = 6x^2 + 3$

- **Solution**:
$$
g'(x) = 12x
$$

## Conclusion

Understanding these basic rules makes differentiating polynomials a straightforward task. Each term in the polynomial is differentiated individually, and the results are summed to find the derivative of the entire polynomial.

# Advanced Polynomial Derivative Examples Using Exponent Rules

Understanding exponent rules can greatly simplify the process of differentiating more complex polynomial expressions. Let's look at some examples, especially focusing on terms like $\frac{1}{x^n}$ which can be rewritten using negative exponents.

## Exponent Rule: $\frac{1}{x^n} = x^{-n}$

When you have a term like $\frac{1}{x^n}$, it can be rewritten as $x^{-n}$. This is useful because it allows us to apply the Power Rule directly.

### Example 1: Differentiate $f(x) = \frac{1}{x^3}$

- **Rewrite the Function**: $f(x) = x^{-3}$
- **Apply the Power Rule**:
$$
f'(x) = -3 \cdot x^{-3-1} = -3x^{-4}
$$
- **Simplify**: $f'(x) = -\dfrac{3}{x^4}$

### Example 2: Differentiate $g(x) = 2x^4 - \frac{5}{x^2}$

- **Rewrite Using Exponent Rules**: $g(x) = 2x^4 - 5x^{-2}$
- **Apply the Power Rule**:
$$
g'(x) = 8x^3 - (-2) \cdot 5x^{-2-1}
$$
$$
= 8x^3 + 10x^{-3}
$$
- **Simplify**: $g'(x) = 8x^3 + \dfrac{10}{x^3}$

### Example 3: Differentiate $h(x) = \dfrac{4}{x} - \dfrac{1}{x^3}$

- **Rewrite Using Exponent Rules**: $h(x) = 4x^{-1} - x^{-3}$
- **Apply the Power Rule**:
$$
h'(x) = -4x^{-1-1} - (-3)x^{-3-1}
$$
$$
= -4x^{-2} + 3x^{-4}
$$
- **Simplify**: $h'(x) = -\dfrac{4}{x^2} + \dfrac{3}{x^4}$

## Conclusion

These examples demonstrate how exponent rules, such as rewriting $\frac{1}{x^n}$ as $x^{-n}$, allow us to apply the Power Rule to a wider range of polynomial expressions. This technique is essential for efficiently differentiating complex polynomial functions.

# Derivatives of Radical Functions: Square Roots, Cube Roots, and Beyond

In calculus, understanding how to differentiate functions involving square roots, cube roots, and other radicals is crucial. These can be approached by rewriting radicals as fractional exponents and then applying the power rule.

## Converting Radicals to Fractional Exponents

A radical expression like $\sqrt[n]{x}$ can be rewritten as $x^{\tfrac{1}{n}}$. This makes it easier to apply the derivative rules.

### Square Roots: $\sqrt{x} = x^{\frac{1}{2}}$

- To differentiate a square root, rewrite $\sqrt{x}$ as $x^{\tfrac{1}{2}}$.

### Cube Roots: $\sqrt[3]{x} = x^{\frac{1}{3}}$

- Similarly, a cube root $\sqrt[3]{x}$ becomes $x^{\tfrac{1}{3}}$.

### General Rule for Radicals

- For any root, $\sqrt[n]{x} = x^{\tfrac{1}{n}}$.

## Examples of Differentiating Radical Functions

### Example 1: Differentiate $f(x) = \sqrt{x}$

- **Rewrite the Function**: $f(x) = x^{\tfrac{1}{2}}$
- **Apply the Power Rule**:
$$
f'(x) = \frac{1}{2} \cdot x^{\tfrac{1}{2} - 1} = \frac{1}{2}x^{-\tfrac{1}{2}}
$$
- **Simplify**: $f'(x) = \dfrac{1}{2\sqrt{x}}$

### Example 2: Differentiate $g(x) = \sqrt[3]{x^2}$

- **Rewrite Using Fractional Exponents**: $g(x) = x^{\tfrac{2}{3}}$
- **Apply the Power Rule**:
$$
g'(x) = \frac{2}{3} \cdot x^{\tfrac{2}{3} - 1} = \frac{2}{3}x^{-\tfrac{1}{3}}
$$
- **Simplify**: $g'(x) = \dfrac{2}{3\sqrt[3]{x}}$

:::spoiler
<summary>Advanced Example Using the Chain Rule from 3.4</summary>

Differentiate $h(x)=\sqrt{x^3+1}$

- **Rewrite the Function**: $h(x) = (x^3 + 1)^{\tfrac{1}{2}}$
- **Apply the Chain Rule and Power Rule**:
$$
h'(x) = \frac{1}{2}(x^3 + 1)^{-\tfrac{1}{2}} \cdot 3x^2
$$
- **Simplify**: $h'(x) = \dfrac{3x^2}{2\sqrt{x^3 + 1}}$
:::
## Conclusion

By converting radicals to fractional exponents, the process of differentiating functions involving square roots, cube roots, and other radicals becomes an application of the familiar power rule, sometimes combined with the chain rule for more complex functions.