The exponential function ex
The number e is defined so that ex is its own derivative - the only function (up to scaling) that doesn’t change when differentiated.
Only when a = e does the slope at x = 0 equal 1 - so ex is the unique exponential equal to its own derivative.
Proof - why is ex its own derivative?
Using the first principles definition:
e is defined as the unique base for which the growth rate at x = 0 equals exactly 1. For any other base b, you would get ln(b) here instead of 1. Since ln(e) = 1, the limit equals 1 only for b = e.
Proof - differentiating ax
Rewrite using e and the natural log. Since :
The chain rule is needed here: outer differentiates to , inner differentiates to .
Worked examples
Practice
The natural logarithm ln(x)
ln(x) is the inverse of ex - it undoes the exponential. Its derivative has a beautifully simple form.
y = ln(x) and y = ex are reflections of each other in y = x. The tangent at x = 1 has slope 1, matching d/dx(ln x) = 1/x at x = 1.
Proof - d/dx(ln x) via implicit differentiation
Let , so . Differentiate both sides with respect to x:
This uses implicit differentiation - covered in the last tab.
The composite rule: d/dx(ln f(x))
Apply the chain rule with , :
In words: the derivative of the inside, over the inside.
Worked examples
Practice
Differentiating sin(x) and cos(x)
Key results - the sine-cosine ladder
Differentiate as you go down
Proof - d/dx(sin(x)) from first principles
Two key limits are needed (proved from the geometry of a unit circle):
Apply first principles with the addition formula :
Proof - d/dx(cos(x))
Using the addition formula :
Worked examples
Practice
The chain rule
Used when one function is composed inside another - f applied to g(x). The rule tells you how to find the derivative of the whole composition.
u₀ is the value the inner function passes to the outer. The red tangent slope on the top graph is g′(x₀); the amber slope on the bottom is f′(u₀). Their product is dy/dx.
- Identify the outer function f and inner function g. Let .
- Differentiate the outer w.r.t. u: find , leaving the inner untouched.
- Differentiate the inner w.r.t. x: find .
- Multiply:
A small change causes a change in u:
That change in u then causes a change in y:
So:
Think of it as two stretch factors chained together: the inner stretches by g′(x) and the outer stretches by f′(u). The combined stretch is their product.
Worked examples
Practice
The product rule
When you differentiate a product of two functions, each factor takes a turn being differentiated while the other stays.
Think of uv as the area of a rectangle. When x changes by dx, both u and v change slightly. The new total area gains two thin strips - one of width du (green) and one of height dv (amber). The tiny corner piece du · dv vanishes as dx → 0:
So
“differential of the first times second, plus differential of the second times first”
Proof - the product rule from first principles
The second line adds and subtracts the same term - a common algebraic trick to split the limit into two recognisable pieces.
Worked examples
Practice
The quotient rule
For differentiating a ratio of two functions. The denominator must be non-zero.
Derivation - quotient from product + chain rule
Write and apply the product rule:
The chain rule gives .
Worked examples
Practice
Derivatives of other trig functions
All four derivatives are obtained by writing tan, sec, cosec and cot in terms of sin and cos, then applying the quotient rule.
| f(x) | f′(x) |
|---|---|
Derivation -
Write and apply the quotient rule:
We used the Pythagorean identity .
Derivation -
Write and apply the quotient rule (or chain rule):
Derivation -
Write and apply the quotient rule:
Derivation -
Write and apply the quotient rule:
Worked examples - with chain rule
Practice
Implicit differentiation
When y is defined implicitly by an equation, differentiate both sides with respect to x. Any term involving y picks up a dy/dx factor via the chain rule.
Treat y as a function of x. By the chain rule:
In general:
Why does differentiating f(y) bring in dy/dx?
When y depends on x, f(y) is a composite function: first apply x → y, then y → f(y). The chain rule says:
Differentiating with respect to y gives f′(y), but since y itself depends on x we must chain in dy/dx. For example:
- State: differentiate with respect to x (differentiate w.r.t. x)
- For each y term, apply the chain rule and multiply by .
- Collect all terms on one side.
- Factorise and solve for .
Worked examples
Practice
Trigonometric identities
Trig identities let you rewrite expressions into a form that is easier to differentiate. Recognising which identity to use is a key skill.
Power-reducing forms (from double angle)
These make it easy to differentiate sin2x and cos2x without using the chain rule directly.