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Derivatives of a Function, Logarithmic differentiation - Coggle Diagram
Derivatives of a Function
Definition
Limit of average rate of change
Instantaneous rate of change
Notation
f'(x)
dy/dx
Df(x)
Rules of Differentiation
Power Rule
d/dx [x^n] = n·x^(n−1)
Product Rule
d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Quotient Rule
d/dx [f(x)/g(x)] = (f'(x)·g(x) − f(x)·g'(x)) / [g(x)]²
Chain Rule
d/dx [f(g(x))] = f'(g(x))·g'(x)
Constant Rule
d/dx [c] = 0
Sum Rule
d/dx [f(x) + g(x)] = f'(x) + g'(x)
Applications
Slope of tangent line
Velocity and acceleration
Optimization problems
Curve sketching
Higher Order Derivatives
Second derivative: f''(x)
Concavity and inflection points
Differentiability
Continuity vs differentiability
Points of non-differentiability
Techniques
Related rates
Implicit differentiation
Logarithmic differentiation