Derivatives in Calculus
The derivative measures the instantaneous rate of change of a function. It represents the slope of the tangent line at any point.
Derivative Rules
- Power Rule: d/dx[x^n] = nx^(n-1)
- Product Rule: d/dx[f·g] = f'g + fg'
- Quotient Rule: d/dx[f/g] = (f'g - fg')/g²
- Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
Applications
Derivatives are used in physics (velocity, acceleration), economics (marginal analysis), and optimization problems.