Differentiation

• The gradient of a curve at any point equals the gradient of the tangent to the curve at that point.
• The derivative (dy/dx or f′(x)) is the gradient function — it gives the gradient at any x-value.
• Power rule: d/dx(xⁿ) = nxⁿ⁻¹ — works for all real powers including negatives and fractions.
• To find the gradient at a point, substitute the x-coordinate into f′(x).
• Tangent at (a, f(a)): use y − f(a) = f′(a)(x − a).
• Normal is perpendicular to tangent: gradient = −1/f′(a).
• Second derivative f′′(x) or d²y/dx² — differentiate the derivative. Gives the rate of change of the gradient.

What does "gradient of a curve" even mean?

A straight line has one fixed gradient — easy. But a curve is always changing direction, so its steepness is different at every single point. We handle this by "zooming in" on the exact point we care about until the curve looks straight — and the gradient of that straight line is the gradient of the curve there.

Formally: the gradient of a curve at a point = the gradient of the tangent to the curve at that point.

What is differentiation?

Differentiation is a mathematical operation that produces the gradient function (derivative) of any function. Feed it a function; it gives you back a new function that tells you the gradient at any x-value.

How is the derivative found — conceptually?

To find the gradient at point P(x, f(x)) on a curve, pick a second nearby point Q(x+h, f(x+h)). The gradient of the chord PQ is:

closer and closer approaches