Calculus: Differentiation, Integration and Applications

Differentiation is the process of finding the derivative of a function. The derivative, denoted as f'(x) or dy/dx, represents the instantaneous rate of change of a function with respect to its independent variable. Geometrically, it gives the gradient of the tangent to the curve at any given point.

These are fundamental rules for differentiating common function types:\n* Power Rule: If f(x) = ax^n, then f'(x) = nax^(n-1).\n* Constant Multiple Rule: If g(x) = cf(x), then g'(x) = cf'(x).\n* Sum/Difference Rule: If h(x) = f(x) ± g(x), then h'(x) = f'(x) ± g'(x).

Used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Alternatively, if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.

Used to differentiate the product of two functions. If y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v'(x).

Used to differentiate the quotient of two functions. If y = u(x)/v(x), then dy/dx = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2.

Key derivatives to memorise:\n* d/dx (e^x) = e^x\n* d/dx (ln(x)) = 1/x (for x > 0)\n* d/dx (sin(x)) = cos(x)\n* d/dx (cos(x)) = -sin(x)\n* d/dx (tan(x)) = sec^2(x)

The second derivative, f''(x) or d^2y/dx^2, is the derivative of the first derivative. It describes the rate of change of the gradient and is used to determine concavity and classify stationary points (local maxima/minima, points of inflection).

Integration is the reverse process of differentiation. It is used to find a function given its derivative. An indefinite integral represents a family of functions, differing by a constant, and is denoted by ∫f(x) dx. The '+ C' (constant of integration) is crucial.

Fundamental rules for integrating common function types:\n* Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)\n* Constant Multiple Rule: ∫cf(x) dx = c∫f(x) dx\n* Sum/Difference Rule: ∫(f(x) ± g(x)) dx = ∫f(x) dx ± ∫g(x) dx

Key integrals to memorise:\n* ∫e^x dx = e^x + C\n* ∫(1/x) dx = ln|x| + C (for x ≠ 0)\n* ∫sin(x) dx = -cos(x) + C\n* ∫cos(x) dx = sin(x) + C

A definite integral ∫_a^b f(x) dx evaluates the accumulation of a function over a specific interval [a, b]. It represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b. The Fundamental Theorem of Calculus states ∫_a^b f(x) dx = F(b) - F(a), where F(x) is an antiderivative of f(x).

Differentiation is used in various practical contexts:\n* Rates of Change: The derivative dy/dx gives the instantaneous rate of change of y with respect to x.\n* Tangents and Normals: The gradient of the tangent to y = f(x) at x=a is f'(a). The normal is perpendicular to the tangent, so its gradient is -1/f'(a).\n* Stationary Points: Points where f'(x) = 0. These can be local maxima, local minima, or horizontal points of inflection. The second derivative test (f''(x)) helps classify them.\n* Optimisation: Finding maximum or minimum values of a quantity by setting the first derivative to zero.

Integration is used in various practical contexts:\n* Area Under a Curve: The definite integral ∫_a^b f(x) dx calculates the signed area between y = f(x) and the x-axis from x = a to x = b. For geometric area, if f(x) is below the x-axis, the integral will be negative, so absolute value or splitting the integral may be needed.\n* Displacement from Velocity: If v(t) is the velocity function, then ∫v(t) dt gives the displacement function. ∫_a^b v(t) dt gives the net displacement from time a to b.