What is differentiation in maths, and how do you differentiate?

A straight line has the same slope everywhere. A curve does not: y = x² is flat at the bottom and gets steeper and steeper. Differentiation gives a formula for the steepness at every point. That formula is called the derivative.

The derivative of y with respect to x is written dy/dx (say "dee y by dee x"). If the function is called f, the derivative is f′(x) ("f dash of x" or "f prime of x"). At any point, it equals the gradient of the tangent, the straight line that just touches the curve there.

Where does it come from? Take two points on the curve, x and x + h, and work out the gradient of the line joining them: (f(x + h) − f(x)) ÷ h. Now make h smaller and smaller. The line turns into the tangent. For y = x², the gradient is ((x + h)² − x²) ÷ h = (2xh + h²) ÷ h = 2x + h, and as h shrinks to 0 this becomes 2x. This is called differentiation from first principles.

Nobody does that every time, because a pattern appears. This is the power rule:

If y = xⁿ, then dy/dx = nxⁿ⁻¹

In words: multiply by the power, then take one off the power. x³ becomes 3x², x⁵ becomes 5x⁴. Three more rules let you differentiate any polynomial:

• A number in front stays: 4x³ becomes 4 × 3x² = 12x².
• A constant on its own disappears: 7 becomes 0, because a flat line has gradient 0.
• Sums are done term by term: x² + 5x − 3 becomes 2x + 5.

The power rule works for any power, including negative ones and fractions. To use it, first rewrite roots and fractions as powers: √x = x^(1/2) and 1/x = x⁻¹.

Why does it matter? A derivative is a rate of change, like the speed of a car (the rate distance changes with time) or how fast a population grows. Setting dy/dx = 0 finds the stationary points of a curve, which are usually its maximums and minimums (occasionally a point of inflection, like x = 0 on y = x³), which is how engineers and businesses find the strongest, cheapest or most profitable design. Going backwards, from a derivative to the original function, is integration.

To tell a maximum from a minimum, differentiate again to get the second derivative, d²y/dx². If it is positive at the stationary point, it is a minimum; if it is negative, it is a maximum.

What is differentiation in maths?#

It is finding the derivative of a function: a formula for its rate of change, or the gradient of its graph, at every point. In biology, "differentiation" means something else: cells becoming specialised, like stem cells turning into nerve or muscle cells. In teaching, it means adapting lessons for different learners.

What are the rules of differentiation?#

Power rule: xⁿ gives nxⁿ⁻¹. Constants give 0 and constant multiples stay. Chain rule: dy/dx = dy/du × du/dx. Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v².

How do you differentiate from first principles?#

Use f′(x) = limit as h → 0 of (f(x + h) − f(x)) ÷ h. Expand f(x + h), cancel, divide by h, then let h become 0. For x³ you get 3x² + 3xh + h², which becomes 3x².

What are the derivatives of sin x, cos x and tan x?#

sin x gives cos x, cos x gives −sin x, and tan x gives sec² x (x in radians). Also sec x gives sec x tan x and cot x gives −cosec² x. For the inverses, sin⁻¹x gives 1/√(1 − x²) and tan⁻¹x gives 1/(1 + x²).

How do you differentiate eˣ, aˣ and ln x?#

eˣ is its own derivative. aˣ gives aˣ ln a, and ln x gives 1/x. With the chain rule, e^(x²) gives 2x e^(x²).

How is differentiation used in physics?#

Velocity is the derivative of displacement with time (v = ds/dt), and acceleration is the derivative of velocity (a = dv/dt). The gradient of a distance–time graph is speed for the same reason.

What is the difference between differentiation and integration?#

They are opposites. Differentiation goes from a quantity to its rate of change (distance to speed); integration goes from a rate of change back to the quantity (speed to distance) and finds areas under curves.