What does differentiation actually find?

The gradient function. Where original function gives a position or value, its derivative gives rate of change at every point.

What is the power rule?

For any power of x, multiply by the power then reduce power by one. So x⁵ differentiates to 5x⁴ and x differentiates to 1.

When do I use the chain rule?

Whenever one function sits inside another, such as (2x + 3)⁵ or sin(3x). Differentiate outer function, keep the inside unchanged, then multiply by the inside's derivative.

Why does a constant differentiate to zero?

A constant never changes, so its rate of change is zero. Graphically it is a horizontal line with zero gradient everywhere.

How do I find maximum and minimum points?

Set first derivative to zero and solve for x, then use second derivative at each solution, negative for a maximum, positive for a minimum.

Specialist Calculator

Differentiation Calculator

Name: Differentiation Calculator
Author: CalZone

Differentiation is tool that turns a curve into information, gradients, rates of change, maximum points and minimum points all fall out of same operation. Our differentiation calculator finds derivative of any function step by step, naming rule applied at each stage so working teaches the method rather than just delivering an answer.

Differentiation Solver

Power Rule: d/dx (axⁿ)

Coefficient (a)

Power (n)

Derivative (dy/dx)6x

The Power Rule

To differentiate a power of x, multiply the expression by the original power (n) and then decrease the power by one. Mathematically: nxⁿ⁻¹.

How the Differentiation Calculator Works

Type your function using standard notation, with ^ for powers and calculator returns derivative along with full working. It handles polynomials, trigonometric, exponential, logarithmic functions and any combination of them and can differentiate again for second and higher derivatives.

The derivative of a function f(x) is written f'(x) or dy/dx and it measures gradient of curve at any point, or equivalently rate at which y changes as x changes. Everything the calculator does is built from a small set of rules applied in the right order.

The Rules of Differentiation

Five rules cover almost every function you will meet:

Power rule: the derivative of xⁿ is n x xⁿ⁻¹
Sum rule: differentiate each term separately
Product rule: for u x v the derivative is u'v + uv'
Quotient rule: for u ÷ v the derivative is (u'v - uv') ÷ v²
Chain rule: for a function of a function, differentiate the outside then multiply by the derivative of the inside

Alongside those sit standard derivatives worth memorising, sin x becomes cos x, cos x becomes -sin x, eˣ stays as eˣ and ln x becomes 1/x. The art of differentiation is not knowing, rules but spotting which one a function needs and calculator's step output labels exactly that decision at every stage.

Worked Example

Differentiate y = 3x⁴ - 5x² + 7x - 2:

3x⁴ becomes 12x³ by the power rule
-5x² becomes -10x
7x becomes 7, since x¹ drops to x⁰ = 1
The constant -2 becomes 0

So dy/dx = 12x³ - 10x + 7. Reading answer, at any value of x, this expression gives exact gradient of original curve, positive where curve climbs and negative where it falls.

Stationary Points: Where Differentiation Earns Its Keep

The most common exam use of a derivative is finding where a curve turns. At a maximum or minimum gradient is zero, so setting dy/dx = 0 and solving locates every stationary point. The second derivative then classifies them, d²y/dx² negative means a maximum, positive means a minimum and zero means further investigation, possibly a point of inflection.

This is also where differentiation leaves classroom. Maximising profit, minimising material in a container design and finding the peak of a projectile's flight are all the same three step routine, differentiate, set to zero, classify.

Differentiation at A Level

Differentiation enters at AS with power rule and stationary points, then A Level adds product, quotient, chain rules, trigonometric and exponential derivatives, implicit, parametric differentiation and rates of change problems. One definition examiners still test directly, differentiation from first principles, where f'(x) is the limit of (f(x+h) - f(x)) ÷ h as h approaches zero. The rules are shortcuts for that limit and knowing the definition is what separates understanding from memorising.

A revision habit that works, differentiate on paper first, then run same function through calculator and compare line by line. A mismatch shows precisely which rule you misapplied, which is far more useful than only checking the final answer.