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A2 Level · Pure Mathematics 3 WMA13

Further Differentiation

Revise Further Differentiation for Pure Mathematics 3 WMA13 (A2 Level) — revision notes and instant AI marking.

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What You'll Learn

By the end of this chapter, you'll be able to differentiate any function — whether it's a product of two functions, a fraction, or a function nested inside another function. You'll master four powerful rules that turn impossible-looking derivatives into manageable steps.

Five key formulas for differentiating standard functions (exponential, logarithmic, trigonometric)
The Chain Rule — for functions inside functions
The Product Rule — for functions multiplied together
The Quotient Rule — for functions divided by each other
Reciprocal and inverse trigonometric derivatives

1. Differentiating Standard Functions

These are the foundation. You need to know them (or know how to derive them quickly):

Power Function

$d/dx (x n) = n\cdotx n-1$

This is the most basic rule. The power comes down, and the exponent decreases by 1.

Exponential Function

$d/dx (e x) = e x$

This is unique — e^x is its own derivative. The exponential function is self-replicating.

General Exponential

$d/dx (a x) = a x \cdotln(a)$ where $a > 0$

The natural logarithm of the base acts as a scaling constant.

Natural Logarithm

$d/dx (ln x) = 1/x$

The derivative of ln is simply the reciprocal of x. Notice it's undefined at x = 0.

Key Insight: ln and e are mathematical inverses — if e is its own derivative, then the derivative of ln is its reciprocal. This isn't coincidence; it's by design.

Trigonometric Functions

Learn these as a block:

$d/dx (sin x) = cos x$

$d/dx (cos x) = -sin x$

$d/dx (tan x) = sec 2 x$

Notice: sine derivative is cosine (positive). Cosine derivative flips sign. Tangent becomes sec-squared.

Special Cases: When the Argument Is Not Just $x$

These look different, but they're all applications of the Chain Rule (coming next):

If $y = e kx$ , then $dy/dx = k\cdote kx$

The constant $k$ from the exponent multiplies the result.

If $y = ln x$ , then $dy/dx = 1/x$

⚠️ This is always 1/x, NOT k/x, even if you have $y = ln(kx)$ . The constant disappears!

If $y = a kx$ , then $dy/dx = k\cdotln(a)\cdota kx$

The constant $k$ multiplies both the ln(a) and the function itself.

Common Mistake: Students often write the derivative of

ln(5x)

5/x

, but it's actually

1/x

. Why? Because

ln(5x) = ln(5) + ln(x)

, and ln(5) is a constant with derivative 0.

Worked Example: Find the derivative of $y = 3e 2x - 5ln x$

Step 1: Differentiate term by term.

Step 2: For

3e 2x

: The constant 3 stays. The exponent 2 comes down. Result:

3 \cdot 2 \cdot e 2x = 6e 2x

Step 3: For

-5ln x

: The constant −5 stays. The derivative of ln x is 1/x. Result:

-5 \cdot (1/x) = -5/x

Answer:

dy/dx = 6e 2x - 5/x

Practice Question 1

Find the derivative of $y = 4e 3x + 7ln x - 2sin x$

2. The Chain Rule

3. The Product Rule

4. The Quotient Rule

What to Memorise

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Also in the full note

5. Reciprocal & Inverse Trigonometric Functions
Exam Tips & Common Pitfalls
The Core Idea
Special Case: Logarithms of Functions
Why the Product Rule?
Reciprocal Trigonometric Functions
Inverse Trigonometric Functions
Core Formulas & Rules

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Further Differentiation