Differentiation of standard functions

Welcome to Differentiation!

Hi there! Today, we are diving into one of the most powerful tools in all of mathematics: Differentiation. If you’ve ever wondered how fast a car is accelerating at a specific split-second, or how to find the exact slope of a curvy hill, differentiation is the answer.

In this chapter, we are going to learn how to differentiate "standard functions." Don't worry if that sounds fancy—it’s mostly just a set of simple rules for dealing with powers of \( x \). Once you learn the "secret move," you’ll be able to solve these problems in your sleep!

1. The Golden Rule: The Power Rule

The core of this chapter is the Power Rule. This is the "standard" way we differentiate any term where \( x \) is raised to a power (like \( x^2 \) or \( x^5 \)).

The Formula:
If \( y = x^n \), then the derivative (the gradient) is:
\( \frac{dy}{dx} = nx^{n-1} \)

How to do it (The "Drop and Chop" Mnemonic):
1. Drop: Take the power (\( n \)) and "drop" it down to the front to multiply.
2. Chop: "Chop" one off the power (subtract 1 from the original power).

Example: Differentiate \( y = x^4 \).
- Drop the 4 to the front: \( 4x \)
- Subtract 1 from the power: \( 4x^3 \)
- So, \( \frac{dy}{dx} = 4x^3 \).

Quick Review: Important Notation

Remember, \( \frac{dy}{dx} \) and \( f'(x) \) mean the exact same thing! They both just mean "the derivative of."

2. Dealing with Coefficients (Constant Multiples)

What if there is already a number in front of the \( x \)? We call this a coefficient or a constant multiple.

The Trick: The number in front just waits there and gets multiplied by the power you "drop" down.

Example: Differentiate \( y = 5x^3 \).
- Drop the 3 down to meet the 5: \( (3 \times 5)x \)
- Subtract 1 from the power: \( 15x^2 \)
- So, \( \frac{dy}{dx} = 15x^2 \).

Did you know?
Differentiation is all about finding the rate of change. In physics, if your function represents distance, differentiating it gives you the speed!

3. Rational Exponents: Negative and Fractional Powers

The syllabus says you need to be able to handle rational values of \( n \). This is just a math-way of saying "fractions and negative numbers." The rule stays exactly the same, but you need to remember your Index Laws from GCSE.

A. Negative Powers (Fractions)

Sometimes \( x \) is on the bottom of a fraction. You must move it up and make the power negative before you differentiate.
Rule: \( \frac{1}{x^n} = x^{-n} \)

Example: Differentiate \( y = \frac{1}{x^2} \).
- Rewrite first: \( y = x^{-2} \)
- Drop the power: \( -2x \)
- Subtract 1: \( -2x^{-3} \) (Careful! \( -2 - 1 = -3 \))
- Final answer: \( \frac{dy}{dx} = -2x^{-3} \) or \( -\frac{2}{x^3} \).

B. Fractional Powers (Roots)

Square roots and cube roots are just hidden fractions.
Rule: \( \sqrt{x} = x^{1/2} \) and \( \sqrt[n]{x^m} = x^{m/n} \)

Example: Differentiate \( y = \sqrt{x} \).
- Rewrite first: \( y = x^{1/2} \)
- Drop the power: \( \frac{1}{2}x \)
- Subtract 1: \( \frac{1}{2}x^{-1/2} \) (Because \( \frac{1}{2} - 1 = -\frac{1}{2} \))
- So, \( f'(x) = \frac{1}{2}x^{-1/2} \).

Don't worry if this seems tricky at first! Working with fractions and negatives is where most students make small mistakes. Just take it one step at a time: Rewrite, then Differentiate.

4. Sums and Differences (The "One by One" Rule)

If you have a long string of terms (a polynomial), just differentiate each part separately. This is called differentiating sums and differences.

Example: Differentiate \( y = x^3 + 4x^2 - 10 \).
- Differentiate \( x^3 \rightarrow 3x^2 \)
- Differentiate \( 4x^2 \rightarrow 8x^1 \) (or just \( 8x \))
- Differentiate \( -10 \rightarrow 0 \) (The "Disappearing Constant" rule!)
- Answer: \( \frac{dy}{dx} = 3x^2 + 8x \).

The "Disappearing Constant" Analogy

Think of a constant number (like 10) as a flat, horizontal line on a graph. A flat line has a gradient of zero. That’s why, when you differentiate a plain number without an \( x \), it simply disappears!

5. Common Mistakes to Avoid

• The "Power Zero" slip-up: When you differentiate \( x \), it becomes 1. Why? Because \( x \) is actually \( x^1 \). Drop the 1, and you get \( 1x^0 \). Anything to the power of 0 is 1. So, the derivative of \( 7x \) is just \( 7 \).
• Forgetting to rewrite roots: Never try to differentiate a root while it's still under the \(\sqrt{}\) sign. Always change it to a power first!
• Negative number subtraction: Remember that \( -1 - 1 = -2 \), not \( 0 \). When power-chopping negative numbers, they look like they are getting "bigger" (e.g., -3 becomes -4).

Key Takeaway Summary

1. Rewrite: Make sure every term looks like \( ax^n \).
2. Drop: Multiply the coefficient by the current power.
3. Chop: Subtract 1 from the power.
4. Constants: Plain numbers (without \( x \)) differentiate to 0.
5. Terms: Treat each part of a plus/minus equation as its own mini-problem.

Quick Review Table:
\( y = k \text{ (number)} \rightarrow \frac{dy}{dx} = 0 \)
\( y = kx \rightarrow \frac{dy}{dx} = k \)
\( y = x^n \rightarrow \frac{dy}{dx} = nx^{n-1} \)

Great job! You now have the fundamental "Standard Function" skills required for the OCR AS Level. Practice a few of these, and you'll be ready for the next step: using these gradients to find tangents and normals!