Derivatives: The Foundation of Calculus
The derivative measures the instantaneous rate of change of a function at any point. If f(x) describes position, then f′(x) describes velocity. The derivative is the slope of the tangent line to the curve, and it is fundamental to physics (motion, forces), engineering (optimization), economics (marginal cost), machine learning (gradient descent), and every quantitative field.
Core Differentiation Rules
Power Rule: d/dx [x^n] = n*x^(n-1)
Constant: d/dx [c] = 0
Constant Multiple: d/dx [c*f] = c*f'
Sum/Diff: d/dx [f +/- g] = f' +/- g'
Product: d/dx [f*g] = f'g + fg'
Quotient: d/dx [f/g] = (f'g - fg') / g^2
Chain: d/dx [f(g(x))] = f'(g(x)) * g'(x)
Trig: sin->cos, cos->-sin, tan->sec^2
Exp/Log: e^x->e^x, ln(x)->1/x
Constant: d/dx [c] = 0
Constant Multiple: d/dx [c*f] = c*f'
Sum/Diff: d/dx [f +/- g] = f' +/- g'
Product: d/dx [f*g] = f'g + fg'
Quotient: d/dx [f/g] = (f'g - fg') / g^2
Chain: d/dx [f(g(x))] = f'(g(x)) * g'(x)
Trig: sin->cos, cos->-sin, tan->sec^2
Exp/Log: e^x->e^x, ln(x)->1/x
Applications
In optimization, setting f′(x)=0 finds maxima and minima. Related rates problems connect changing quantities. Linear approximation uses f(x+h) ≈ f(x)+f′(x)*h. In physics, derivatives connect position, velocity, and acceleration.
Key Patterns
Polynomials: bring down power, reduce by 1. Trig: sin becomes cos, cos becomes -sin. Exponentials: e^x is its own derivative. Logarithms: ln(x) becomes 1/x. Chain rule: multiply by derivative of the inside.