What is Differentiation?
Differentiation is one of the two fundamental operations of calculus (the other being integration). At its core, differentiation answers the question: how fast is this quantity changing? More precisely, it finds the instantaneous rate of change of a function at any given point — which is the same as the gradient (slope) of the curve at that point.
If you have a curve y = f(x), the derivative f'(x) — also written dy/dx — gives you the gradient of the tangent line to that curve at every value of x. When the curve is rising steeply, dy/dx is large and positive. When it is flat (at a maximum or minimum), dy/dx equals zero. When it slopes downward, dy/dx is negative.
Differentiation was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Newton used dot notation (ẋ) mainly for physics, while Leibniz introduced the dy/dx notation used in mathematics and engineering today.
dy/dx Notation Explained
The notation dy/dx (read "dee y by dee x") means the derivative of y with respect to x. The "d" represents an infinitesimally small change. So dy/dx is the ratio of an infinitesimally small change in y to an infinitesimally small change in x — which is precisely the gradient of the curve.
Alternative notations for the derivative of f(x) include:
- f'(x) — Lagrange or prime notation (most common in pure maths)
- dy/dx — Leibniz notation (most common in applied maths and physics)
- Df(x) — Euler's operator notation
- ẋ — Newton's dot notation (mainly used in mechanics for time derivatives)
In A-Level mathematics, you will mostly encounter f'(x) and dy/dx. For the second derivative (the rate of change of the rate of change), you write f''(x) or d²y/dx².
The Power Rule: Differentiating x^n
The power rule is the most frequently used rule in differentiation and applies to any power of x:
With constant: If f(x) = ax^n, then f'(x) = an · x^(n-1)
The steps are always: (1) bring the power down as a multiplier, (2) reduce the power by one. Examples:
- d/dx(x³) = 3x² — power 3 comes down, new power is 2
- d/dx(4x&sup5;) = 20x&sup4; — 4 times 5 = 20, new power is 4
- d/dx(x) = 1 — because x = x¹, so 1·x&sup0; = 1
- d/dx(7) = 0 — any constant differentiates to zero
- d/dx(x^(-2)) = -2x^(-3) = -2/x³ — works for negative powers too
- d/dx(√x) = d/dx(x^(1/2)) = (1/2)x^(-1/2) = 1/(2√x)
Worked Example: Differentiate f(x) = x³ + 2x² − 5x + 3
Apply the power rule to each term separately (sum rule):
Therefore: f'(x) = 3x² + 4x − 5
To find the gradient at x = 2: f'(2) = 3(4) + 4(2) − 5 = 12 + 8 − 5 = 15
The Chain Rule
The chain rule is used to differentiate composite functions — functions of functions. If y = f(g(x)), where g(x) is the "inner" function and f is the "outer" function:
A practical way to remember it: differentiate the outer function (leaving the inner unchanged), then multiply by the derivative of the inner function.
Worked Example: Differentiate y = sin(3x²)
Outer function: sin(•) → derivative cos(•). Inner function: 3x² → derivative 6x.
Worked Example: Differentiate y = (2x + 1)&sup5;
Outer: (•)&sup5; → 5(•)&sup4;. Inner: (2x + 1) → 2.
The Product Rule
When two functions are multiplied together, use the product rule. If y = u(x) · v(x):
Memory aid: "first times derivative of second, plus second times derivative of first"
Worked Example: Differentiate y = x² · sin(x)
Let u = x² (so u' = 2x) and v = sin(x) (so v' = cos(x)).
The Quotient Rule
When one function is divided by another, use the quotient rule. If y = f(x) / g(x):
Memory aid: "low d-high minus high d-low, square the bottom and away we go"
Worked Example: Differentiate y = sin(x) / x
f(x) = sin(x), f'(x) = cos(x). g(x) = x, g'(x) = 1.
Partial Differentiation Explained
When a function depends on more than one variable — for example f(x, y) or f(x, y, z) — we can differentiate with respect to each variable separately. This is called partial differentiation.
The notation changes from d to the curly ∂ symbol: ∂f/∂x means "the partial derivative of f with respect to x" — treating all other variables as constants.
Worked Example: Partial derivatives of f(x,y) = x²y + 3xy² + 2y
∂f/∂x (treat y as constant):
∂f/∂y (treat x as constant):
Note: The term 2y differentiates to 0 when finding ∂f/∂x (because y is constant), but differentiates to 2 when finding ∂f/∂y.
Partial derivatives are essential in multivariable calculus, thermodynamics, economics (marginal analysis), and machine learning (gradient descent uses partial derivatives to minimise loss functions).
Differentiation in A-Level Maths
In A-Level Mathematics (both AQA and Edexcel), differentiation typically appears in:
- C1/AS Pure: Power rule, differentiating polynomials, finding gradients and tangents
- C2/A2 Pure: Chain rule, product rule, quotient rule, differentiating trig and exponentials
- A2 Pure: Implicit differentiation, parametric differentiation, second derivatives
- Mechanics: Velocity (dx/dt) and acceleration (d²x/dt²) as derivatives of position
Differentiation also appears in the applications of calculus: finding stationary points (where f'(x) = 0), determining whether they are maxima or minima (using f''(x) — positive means minimum, negative means maximum), and curve sketching.
Applications of Differentiation
Finding maximum and minimum points: Set f'(x) = 0 and solve for x. These are the stationary points. Check f''(x): if f''(x) < 0 it is a maximum, if f''(x) > 0 it is a minimum.
Velocity and acceleration: If s(t) is displacement as a function of time, then velocity v = ds/dt and acceleration a = dv/dt = d²s/dt².
Optimisation in business: If a company's profit P depends on output q, then dP/dq = 0 gives the profit-maximising output. This is marginal profit = 0.
Rates of change in science: In chemistry, reaction rates are derivatives of concentration with respect to time. In physics, the rate of heat flow is a partial derivative of temperature with respect to position.
Machine learning: Neural networks are trained using gradient descent, which repeatedly computes partial derivatives (the gradient) of the loss function with respect to all weights and biases.
Frequently Asked Questions
What is dy/dx? +
dy/dx is Leibniz notation for the derivative of y with respect to x. It represents the instantaneous rate of change of y as x changes — equivalently, the gradient of the curve y = f(x) at any given point. If y = x², then dy/dx = 2x, meaning at x = 3 the gradient of the parabola is 6.
How do you differentiate x^n? +
Use the power rule: multiply by the power, then reduce the power by one. d/dx(x^n) = n·x^(n-1). For example: d/dx(x&sup5;) = 5x&sup4;, d/dx(x²) = 2x, d/dx(x) = 1, d/dx(7) = 0. This works for any real power, including negative and fractional powers.
What is partial differentiation? +
Partial differentiation finds the derivative of a multivariable function with respect to one variable, treating all other variables as constants. The notation uses ∂ instead of d. For f(x,y) = x²y + 3y: ∂f/∂x = 2xy (y is constant, 3y drops out) and ∂f/∂y = x² + 3 (x² is the coefficient of y).
How do you use the chain rule? +
The chain rule applies to composite functions f(g(x)): d/dx[f(g(x))] = f'(g(x)) × g'(x). Step 1: differentiate the outer function (leaving the inner function unchanged). Step 2: multiply by the derivative of the inner function. Example: d/dx[e^(x²)] = e^(x²) × 2x = 2x·e^(x²).
What is the derivation of differentiation from first principles? +
From first principles, the derivative is defined as: f'(x) = lim(h→0) [f(x+h) − f(x)] / h. For f(x) = x²: [(x+h)² − x²] / h = [x² + 2xh + h² − x²] / h = [2xh + h²] / h = 2x + h. As h→0, this gives 2x, confirming the power rule.
What is the derivative of x/(1+x)? +
Use the quotient rule with f = x (f' = 1) and g = 1+x (g' = 1). d/dx[x/(1+x)] = [(1)(1+x) − (x)(1)] / (1+x)² = [1+x − x] / (1+x)² = 1/(1+x)². This result is useful in logistic functions and economics.