What is differentiation?
Answer
Differentiation is the process of finding the derivative of a function, which measures its instantaneous rate of change with respect to a variable.
The working
Core idea
Differentiation gives the rate at which one quantity changes relative to another. Geometrically, the derivative at a point equals the slope of the tangent to the curve there.
Definition by limit
The derivative is defined as f'(x) = lim(h→0) [f(x + h) − f(x)] ÷ h. This first-principles form is the foundation of all differentiation rules.
Notation
The derivative of y = f(x) is written as dy/dx or f'(x). It represents how y changes for a tiny change in x.
Power rule example
For y = xⁿ, the derivative is dy/dx = n·xⁿ⁻¹. So if y = x³, then dy/dx = 3x².
Where it is used
It finds slopes, velocities, and maxima or minima. For example, differentiating a distance function gives velocity.
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