Introduction to Calculus
Understand limits, differentiation from first principles, and apply basic differentiation rules to find gradients and rates of change.
Limits
A limit describes the value a function approaches as the input approaches a particular value. We write:
limx→a f(x) = L
As x gets closer and closer to a, f(x) gets closer and closer to L.
Example: limx→2 (x2 − 4)/(x − 2)
Direct substitution: 0/0 — indeterminate form!
Factorise: (x2 − 4)/(x − 2) = (x+2)(x−2)/(x−2) = x + 2 (for x ≠ 2)
Now substitute: limx→2 (x + 2) = 4
Differentiation from First Principles
The derivative of a function at a point gives the gradient (slope) of the tangent at that point. From first principles:
First Principles Definition
f′(x) = limh→0 f(x + h) − f(x) h
Example: Differentiate f(x) = x2 from first principles
f(x+h) = (x+h)2 = x2 + 2xh + h2
f(x+h) − f(x) = 2xh + h2 = h(2x + h)
[f(x+h) − f(x)] / h = 2x + h
limh→0 (2x + h) = 2x
So f′(x) = 2x
Differentiation Rules
Instead of using first principles every time, we use standard differentiation rules:
Power Rule
If f(x) = xn, then f′(x) = nxn−1
Constant Multiple
If f(x) = cf(x), then f′(x) = c × f′(x)
Sum Rule
(f + g)′ = f′ + g′
Constant Rule
If f(x) = c, then f′(x) = 0
Example: Differentiate f(x) = 3x4 − 5x2 + 7x − 2
f′(x) = 3(4x3) − 5(2x) + 7(1) − 0
f′(x) = 12x3 − 10x + 7
Knowledge Check
Test your understanding of introductory calculus concepts.
Question 1
Evaluate: limx→3 (2x + 1)
Question 2
Differentiate using the power rule: f(x) = x5
Question 3
What is the derivative of f(x) = 7 (a constant)?
Question 4
Evaluate: limx→1 (x2 − 1)/(x − 1)
Question 5
Differentiate: f(x) = 4x3 − 2x + 9
Question 6
Find the gradient of f(x) = x2 − 3x at x = 4.
Question 7
What does the derivative f′(x) represent geometrically?
Question 8
Differentiate: f(x) = 6√x (hint: rewrite as a power of x first)
Question 9
A particle moves with displacement s(t) = t3 − 6t2 + 9t. Find the velocity at t = 2.
Question 10
Using first principles, find f′(x) for f(x) = 3x + 2.
Key Concepts Summary
- ●A limit describes the value a function approaches as the input approaches a given value.
- ●First principles: f′(x) = limh→0 [f(x+h) − f(x)] / h.
- ●Power rule: d/dx(xn) = nxn−1.
- ●The derivative gives the gradient of the tangent and the instantaneous rate of change.
- ●Constants differentiate to 0; sums are differentiated term by term.