Introduction to Differentiation and Basic Rules

TL;DR

Differentiation is all about finding the instantaneous rate of change of a function, like the speed of a moving object at an exact moment. It helps us understand how quickly one quantity changes in response to another. We'll start by looking at basic rules for common functions to get you started.

1. The Mental Model

Think of differentiation as finding the slope of a curve at a single point. If you're walking up a hill, differentiation tells you exactly how steep the hill is right where you're standing, not just the average steepness over a long stretch.

2. The Core Material

Differentiation gives us a new function, called the derivative, that tells us the rate of change of the original function at any point. We often use the notation $f'(x)$ (read as "f prime of x") or $\frac{dy}{dx}$ (read as "dee y dee x") for the derivative.

The Power Rule

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This is your most fundamental rule. It helps you differentiate functions that look like $x^n$.

If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

How it works: You bring the power ($n$) down as a multiplier, and then you subtract 1 from the original power.

Examples:
* If $f(x) = x^3$, then $f'(x) = 3x^{3-1} = 3x^2$.
* If $f(x) = x^5$, then $f'(x) = 5x^{5-1} = 5x^4$.
* If $f(x) = x$ (which is $x^1$), then $f'(x) = 1x^{1-1} = 1x^0 = 1$. (Remember $x^0 = 1$)
* If $f(x) = \sqrt{x}$ (which is $x^{1/2}$), then $f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$.

Constant Rule

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If your function is just a constant number, its rate of change is zero. It's not changing!

If $f(x) = c$ (where $c$ is any constant number), then $f'(x) = 0$.

Examples:
* If $f(x) = 7$, then $f'(x) = 0$.
* If $f(x) = -100$, then $f'(x) = 0$.

Constant Multiple Rule

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If you have a constant multiplied by a function, you can just pull the constant out and differentiate the function.

If $f(x) = c \cdot g(x)$, then $f'(x) = c \cdot g'(x)$.

Examples:
* If $f(x) = 5x^3$, we know from the power rule that the derivative of $x^3$ is $3x^2$. So, $f'(x) = 5 \cdot (3x^2) = 15x^2$.
* If $f(x) = -2x^4$, then $f'(x) = -2 \cdot (4x^3) = -8x^3$.

Sum and Difference Rule

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When you have a function that's made up of several terms added or subtracted together, you can differentiate each term separately.

If $f(x) = g(x) \pm h(x)$, then $f'(x) = g'(x) \pm h'(x)$.

Example:
* If $f(x) = 3x^2 + 7x - 5$:
* Derivative of $3x^2$ is $3 \cdot (2x^1) = 6x$.
* Derivative of $7x$ is $7 \cdot (1) = 7$.
* Derivative of $-5$ is $0$.
* So, $f'(x) = 6x + 7 - 0 = 6x + 7$.

Here's how these basic rules fit together:

graph TD
    A["Start Differentiation"] --> B{"Is it a Constant?"}
    B -- Yes --> C["Result: 0 (Constant Rule)"]
    B -- No --> D{"Is it a sum/difference of terms?"}
    D -- Yes --> E["Differentiate Each Term Separately (Sum/Difference Rule)"]
    E --> F{"Is a term a constant times a function?"}
    F -- Yes --> G["Factor out constant, then differentiate function (Constant Multiple Rule)"]
    F -- No --> H{"Is a term x^n?"}
    H -- Yes --> I["Apply Power Rule (nx^(n-1))"]
    H -- No --> J["(Other Rules for more complex functions)"]
    G --> I
    I --> K["Combine results"]
    C --> L["End"]
    K --> L

3. Worked Example

Let's differentiate the function $f(x) = 4x^3 - 2x^2 + \frac{1}{x} + 9$.

First, rewrite any terms so they look like $x^n$. The term $\frac{1}{x}$ can be written as $x^{-1}$.

So, $f(x) = 4x^3 - 2x^2 + x^{-1} + 9$.

Now, let's differentiate each term using the rules:

1. Term 1: $4x^3$

   • Apply Constant Multiple Rule and Power Rule: $4 \cdot (3x^{3-1}) = 4 \cdot (3x^2) = 12x^2$.

2. Term 2: $-2x^2$

   • Apply Constant Multiple Rule and Power Rule: $-2 \cdot (2x^{2-1}) = -2 \cdot (2x^1) = -4x$.

3. Term 3: $x^{-1}$

   • Apply Power Rule: $-1x^{-1-1} = -1x^{-2} = -\frac{1}{x^2}$.

4. Term 4: $9$

   • Apply Constant Rule: $0$.

Finally, combine the derivatives of each term using the Sum and Difference Rule:

$f'(x) = 12x^2 - 4x - \frac{1}{x^2} + 0$
$f'(x) = 12x^2 - 4x - \frac{1}{x^2}$

4. Key Takeaways

• Differentiation finds the instantaneous rate of change or the slope of a tangent line at a point on a curve.
• The Power Rule ($nx^{n-1}$) is essential for differentiating terms like $x^n$.
• The derivative of any constant is always zero.
• Constants multiplied by a function remain as multipliers in the derivative.
• You can differentiate sums and differences of terms one by one.
• Rewrite fractions like $\frac{1}{x^n}$ as $x^{-n}$ before applying the Power Rule.
• Square roots like $\sqrt{x}$ should be written as $x^{1/2}$ for differentiation.

Common Mistakes to Avoid:
- Forgetting to subtract 1 from the power in the Power Rule.
- Differentiating a constant to be 1 instead of 0.
- Applying the Power Rule to a constant (e.g., differentiating 5 to get $5x^0$).
- Not rewriting terms like $\frac{1}{x}$ or $\sqrt{x}$ into $x^n$ form before starting.

5. Now Try It

Take a piece of paper and differentiate the function $g(x) = 6x^4 + 3x - \frac{5}{x^2} + 12$. Work through each term systematically using the rules we covered.

What success looks like: Your final derivative, $g'(x)$, should be an expression with $x$ terms and constants, where each term has been correctly differentiated. Make sure all powers are handled correctly, especially for the $\frac{5}{x^2}$ term.