Use standard differentiation notation including first and second derivatives

What is a Derivative?

Think of a road that goes up and down. The derivative tells you how steep the road is at any point. In maths, it’s the slope of the tangent line to a curve at a specific point.

Notation: If you have a function f(x), its first derivative is written as f'(x) or dy/dx when y = f(x).

Example: For f(x) = x², the slope at x = 3 is f'(3) = 6.

First Derivative Rules

• Power Rule: f(x)=xⁿ ⇒ f'(x)=n·xⁿ⁻¹
  Example: f(x)=x⁴ ⇒ f'(x)=4x³
• Product Rule: (uv)' = u'v + uv'
  Example: f(x)=x·eˣ ⇒ f'(x)=eˣ + x·eˣ = eˣ(1+x)
• Quotient Rule: (u/v)' = (u'v - uv')/v²
  Example: f(x)=x/(x+1) ⇒ f'(x)=1/(x+1)²
• Chain Rule: (g∘h)' = g'(h(x))·h'(x)
  Example: f(x)=sin(x²) ⇒ f'(x)=cos(x²)·2x = 2x·cos(x²)

Example Table: Differentiating Common Functions

Second Derivative f''(x)

The second derivative tells you how the slope itself is changing – it’s the “acceleration” of the function.

Example: For f(x)=x⁴, first derivative is f'(x)=4x³. Differentiating again gives f''(x)=12x².

Interpretation: If f''(x) > 0 the function is concave up (like a cup). If f''(x) < 0 it’s concave down (like a frown).
Use this to locate inflection points where the concavity changes.

📌 Examination Tips

• Always simplify the function first before differentiating.
• Check the domain of the function – derivatives may not exist at points where the function is undefined.
• Use the chain rule for composite functions; remember to differentiate the outer function first, then multiply by the derivative of the inner function.
• When asked for critical points, set f'(x)=0 and solve for x.
• For second derivative tests, evaluate f''(x) at the critical points to determine maxima/minima.
• Write down the steps clearly – examiners look for a logical progression.
• Use colour coding or bold to highlight key results in your handwritten notes.