This mini-lesson covers SL Calculus: the derivative as a gradient, the power rule, tangents & normals, stationary points and optimisation, integration as the reverse of differentiation, definite integrals and area, and kinematics.
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The derivative f′(x) is the gradient of the tangent to y = f(x) — the instantaneous rate of change. It is defined as a limit of gradients of chords.
For each power, bring the index down and reduce it by one; differentiate term by term.
SL standards: also d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.
The gradient of the tangent at x = a is f′(a); the normal is perpendicular, with gradient −1/f′(a).
y = x² at x = 3: f′(x) = 2x so the tangent gradient is 2(3) = 6. Tangent line: y − 9 = 6(x − 3).
f is increasing where f′(x) > 0 and decreasing where f′(x) < 0. A stationary point has f′(x) = 0.
Classify: use the second derivative — f″ > 0 gives a local minimum, f″ < 0 a local maximum. For y = x² − 6x + 5, f′ = 2x − 6 = 0 at x = 3.
Tap a task, then the tool that produces it.
To optimise, write the quantity as a function of one variable, differentiate, set the derivative to zero, and test the nature of the stationary point.
Always check the endpoints and the context (a length or area cannot be negative).
Integration reverses differentiation. Raise the index by one and divide:
Tap a statement on the left, then its matching partner on the right.
A definite integral gives the (signed) area between the curve and the x-axis:
Kinematics: velocity v = ds/dt and acceleration a = dv/dt; going back, s = ∫v dt.
"Gradient / rate of change / tangent" → differentiate. "Area / total / displacement from velocity" → integrate. "Maximum / minimum" → set f′(x) = 0.
Exam habit: never forget +C for indefinite integrals, and substitute limits in the order F(b) − F(a).
Derivative: f′(x) = gradient; power rule d/dx xⁿ = nxⁿ⁻¹
Tangents: tangent gradient f′(a); normal gradient −1/f′(a)
Stationary: f′(x) = 0; classify with f″
Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Definite/kinematics: ∫ₐᵇ = F(b)−F(a); v = ds/dt, a = dv/dt
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